Shashi Bhushan Tiwari
 
IIT-JEE ADVANCED
VOLUME I
5
 
In the past decade and a half, the entrance exam for IITs has seen many changes – in structure as well as in design of the question paper. No doubt, it has become more challenging. It requires high level of conceptual clarity and analytical skill, besides promptness and comprehension ability to excel in this exam. There are frequent surprises in terms of problems which require mathematical rigor or in depth understanding of physical conditions.This book is being presented with a very simple objective – it will test you and nurture you on all parameters which are required to excel in JEE exam. Every chapter in the book has been divided into three sections – LEVEL 1 – This section will test you on all basic fundamentals of the chapter. Problems are not very rigorous though they may be very conceptual. LEVEL 2 – This section will develop all necessary skills required to score a high rank in JEE exam. Few problems in this section may appear lengthy but they are the ones which test your confidence and patience. Don’t be scared of them. LEVEL 3 – This section contains problems that may require exceptional reasoning skill or mathematical ability. Since difficulty level is quite subjective and may vary from person to person — few problems may appear to you as misplaced in three sections described above. I have judged them to the best of my ability besides taking help from some very bright minds.I have not tried to include every other problem that is available in this universe. Most of the books available in market have this issue – in the name of being exhaustive, they have become repetitive. Believe me, while solving problems from this book you will not feel like wasting your time in doing similar problems again and again..Most of the solutions are quite descriptive so that a serious student can understand on his/her own. Diagrams have been included wherever possible to make things lucid.JEE exam being objective, one may challenge the sanctity of a subjective book. Have no doubts in your mind — pattern of a question paper or type of question will never deter you if you have sound grasp of the subject and have developed right kind of temperament. Physics as a subject is notorious and can be learned only by subjecting yourself to the true rigor and complexity. While doing a subjective problem you cannot make a guess and bluff yourself!This collection of problems will appear to you as fresh and challenging. Start and enjoy learning physics! Suggestions are welcome.
 PREFACE
5
 
Preface v
Chapter 1 Basic Maths 1.1 – 1.12Chapter 2 Kinematics 2.1 – 2.101Chapter 3 Newton's Laws 3.1 – 3.134Chapter 4 Work - Power - Energy 4.1 – 4.69Chapter 5 Momentum and Center Of Mass 5.1 – 5.87Chapter 6 Rotational Motion 6.1 – 6.133Chapter 7 Gravitation 7.1 – 7.40Chapter 8 Fluids 8.1 – 8.52Chapter 9 Surface Tension 9.1 – 9.19Chapter 10 Viscosity 10.1 – 10.8Chapter 11 Elasticity 11.1 – 11.9Chapter 12 Simple Harmonic Motion 12.1 – 12.48Chapter 13 Wave Motion 13.1 – 13.47
CONTENTS
4
4
 
L 
EVEL 
 1
 Q.1 In an experiment mileage of a car was measured to be 24
kmpl
(Kilometer per liter of fuel consumed). After the experiment it was found that 4 % of the fuel used during the experiment was leaked through a small hole in the tank. Calculate the actual mileage of the car after the tank was repaired.
 Q.2 A man is standing at a distance of 500m from a building. He notes that angle of elevation of the top of the building is 3.6°. Find the height of the building. Neglect the height of the man and take = 3.14. Q.3 A Smuggler in a hindi film is running with a bag 0.3 m × 0.2 m × 0.2 min dimension. The bag is supposed to be completely filled with gold. Do you think than the director of the film made a technical mistake there? Density of gold is 19.6 g/cc.
 
Q.4
 
A particle moves along the curve 6y = x3+2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.
 
Q.5
 
The area of a regular octagon of side length
a
 is A. 
(a)
Find the time rate of change of area of the octagon if its side length is being increased at a constant rate of
b
m/s
. Is the time rate of change of area of the octagon constant with time? 
(b)
Find the approximate change in area of the octagon as the side length is increased from 2.0 m to 2.001 m.
L 
EVEL 
 2
 Q.6 Spirit in a bowl evaporates at a rate that is proportional to the surface area of the liquid.
 BASIC MATHS
Initially, the height of liquid in the bowl is
 H 
0
. It becomes
 H 
0
2 in time
t 
0
. How much more time will be needed for the height of liquid to become
 H 
0
4. Q.7 Show that the volume of a segment of height h of a sphere of radius R is
V h R h
=
 ( )
133
2
π 
–
hR
 Q.8 The amount of energy a car expends against air resistance is approximately given by E = K ADv2 where E is measured in Joules. K is a constant, A is the cross-sectional area of the car viewed from the front (in m2), D is the distance traveled (in m), and v is the speed of the car (in m/s). Julie wants to drive from Mumbai to Delhi and get good fuel mileage. For the following questions, assume that the energy loss is due solely to air resistance.
 (a)
 Julie usually drives at a speed of 54 Km/hr. How much more energy will she use if she drives 20% faster?
(b)
Harshit drives a very large SUV car, and Julie drives a small car. Every linear dimension of Harshit’s car is double that of Julie’s car. Find the ratio of energy spent by Harshit’s car to
01
4
 
 1.2
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
Julie’s car when they cover same distance. Speed of Harshit was 10% faster compared to Julie’s car. 
(c)
Write the dimensional formula for K. Will you believe that K depends on density of air? Q.9 The volume flow rate
 Q (in m
3
 s
 –1
)
 of a liquid through pipe having diameter d is related to viscosity of water ‘
h
’ (
unit Pascal.
 
s
) and the pressure gradient along the pipe
dPdx 
 [pressure gradient
dPdx 
 is rate of change of pressure per unit length along the pipe], by a formula of the form 
Q k d  dPdx 
a bc
=     
η
Where K is a dimensionless constant. Find a,b and c. Q.10. The potential energy (U) of a particle can be expressed in certain case as
 U  Amr  BMmr 
=
22
2–Where m and M are mass and r is distance. Find the dimensional formulae for constants. Q.11. In the following expression V and g are speed and acceleration respectively. Find the dimensional formulae of
a
 and
b
 
VdV g bV a
–
2
 =
∫ 
 Q.12 The maximum height of a mountain on earth is limited by the rock flowing under the enormous weight above it. Studies show that maximum height depends on young’s modulus (Y) of the rod, acceleration due to gravity (g) and the density of the rock (d). (a) Write an equation showing the dependence of maximum height (h) of mountain on Y, g and d. It is given that unit of Y is Nm
–2
. (b) Take
d 
 = 3 × 10
3
 kg
m
–3
,
Y 
 = 1 × 10
10
 Nm
–2
and
 g
 = 10 ms
–2
and assume that maximum height of a mountain on the surface of earth is limited to 10 km [height of mount Everest is nearly 8 km]. Write the formula for h. Q.13 A particle of mass m is given an initial speed
V 
0
. It experiences a retarding force that is proportional to the speed of the particle (
F 
 =
aV 
).
a
 is a constant. (a) Write the dimensional formula of constant
a
. (b) Using dimensional analysis, derive a formula for stopping time (t) of the particle. Does your formula tell you how ‘t’ depends on initial speed
V 
0
? What can you predict about the constant obtained in the formula? Q.14 Assume that maximum mass m
1
 of a boulder swept along by a river, depends on the speed
V 
 of the river, the acceleration due to gravity g, and the density
d 
 of the boulder. Calculate the percentage change in maximum mass of the boulder that can be swept by the river, when speed of the river increases by 1%. Q.15 A massive object in space causes gravitational lensing. Light from a distant source gets deflected by a massive lensing object. This was first observed in 1919 and supported Einstein’s general theory of relativity. The angle by which light gets deflected due to a massive body depends on the mass (M) of the body, universal gravitational constant (G), speed of light (c) and the least distance (r) between the lensing object and the apparent path of light. Derive a formula for using method of dimensions. Make suitable assumptions. Q.16 The Casimir effect describes the attraction between two unchanged conducting plates placed parallel to each other in vacuum. The astonishing force ( predicted in 1948 by Hendrik Casimir) per unit area of each plate depends on the planck’s constant (h), speed of light (c) and separation between the plates (r). (a) Using dimensional analysis prove that the formula for the Casimir force per unit area on the plates is given by 
F k  hcr 
=
4
where k is a dimensionless constant (b) If the force acting on 1x1 cm plates separated by 1
 m
 is 0.013 dyne, calculate the value of constant k. Q.17. Scattering of light is a process of absorption and prompt re-emission of light by atoms and molecules. Scattering involving particles smaller than wavelength (
l)
of light is known as Rayleigh scattering. Let
a
i
 
be amplitude of incident light on a scatterer of volume V. The scattered amplitude at a distance r from the scatterer is
a
s
. Assume and
a
s
 
a
a
i
 ,
a
s
 1
r 
 and
a
s
 
a 
V. (i) Find the dimensions of the proportionality constant occurring in the expression of
a
s
5
 
 B
 ASIC
 M
 ATHEMATICS
 A
ND
 D
IMENSIONAL
 A
NALYSIS
 
1.3
 (ii) Assuming that this constant depends on
l 
, find the dependence of ratio
aa
si
on
l 
. (iii) Knowing that intensify of light I
a
a
2
 find the dependence of
 I  I 
si
on
l 
. Q.18 It is given that
dx  x  x c
1
21
+ = +
∫
 −
tan. Using methods of dimensions find
dx a x 
22
+
∫ 
 Q.19
BA ABC
 Two point sources of light are fixed at the centre (A) and circumference (point B) of a rotating turn table. A photograph of the rotating table is taken. On the photograph a point A and an arc BC appear. The angle
q 
was measured to be
q
= 10.8°
±
0.1°and the angular speed of the turntable was measured to be
w 
 = (33.3 ± 0.1)revolution per minute. Calculate the exposure time of the camera. Q.20 The speed (V) of wave on surface of water is given by 
V  a b
= +
l p p rl 
22where
l 
is the wavelength of the wave and
r
is density of water.
a
 is a constant and b is a quantity that changes with liquid temperature. (a) Find the dimensional formulae for
a
 and b. (b) Surface wave of wavelength 30 mm have a speed of 0.240
ms
–1
. If the temperature of water changes by 50°
C 
, the speed of waves for same wavelength changes to 0.230
ms
–1
. Assuming that the density of water remains constant at 1 × 10
3
 kg m
–3
, estimate the change in value of ‘b’ for temperature change of 50º
C 
. Q.21 The line of sight of the brightest star in the sky, Sirius has a maximum parallax angle of 
d 
 = 0.74 ± 0.02 arc second when observed at six month interval. The distance between two positions of earth (at six – month interval) is
r 
 = 3.000 × 10
11
m
StarrSunEarthEarth
 Calculate the distance of Sirius from the Sun with uncertainty, in unit of light year. Given 1 ly = 9.460x10
15
 m. ;
p
= 3.14
L 
EVEL 
 3
 
Q.22 You inhale about 0.5 liter of air in each breath and breath once in every five seconds. Air has about 1% argon. Mass of each air particle can be assumed to be nearly 5 × 10
–26
 
kg
. Atmosphere can be assumed to be around 20
km
 thick having a uniform density of 1.2
kg m
–3
. Radius of the earth is
 R
 = 6.4 × 10
6
 m. Assume that when a person breathes, half of the argon atoms in each breath have never been in that person’s lungs before. Argon atoms remain in atmosphere for long-long time without reacting with any other substance. Given : one year = 3.2 × 10
7
s (a) Estimate the number of argon atoms that passed through Newton’s lungs in his 84 years of life. (b) Estimate the total number of argon atoms in the Earth’s atmosphere. (c) Assume that the argon atoms breathed by Newton is now mixed uniformly through the atmosphere, estimate the number of argon atoms in each of your breath that were once in Newton’s lungs. Q.23 A rope is tightly wound along the equator of a large sphere of radius R. The length of the rope is increased by a small amount
 (<< R) and it is pulled away from the surface at a point to make it taut. To what height (h) from the surface will the point rise ? If the radius of the earth is R=6400 km and
 
=
4
 
 1.4
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
ANSWERS
1.
 25 kmpl
2.
 31.40 m
3.
 Yes.
4.
(– 4,
-
313), (4, 11)
5.
(a) 4 (2 + 1)
a
b 
. No, it is not a constant (b)
 
0.0019
m
 2
6.
 
t 
0
2
8.
 (a) 44% higher (b) 4.84 (c) [
 ML
–3
]; Yes
9.
 
a
 = – 1;
b
 = 4;
c
 = 1
10.
 [A] = [
 M 
1
 L
2
T 
–1
][B] = [
 M 
–1
 L
3
T 
–2
]
11.
 [a] =
 L
; [b] =
 L
–1
12.
 (a)
h
 =
k 
 
Y gd 
     
 ;
k 
 =
a const 
 (b)
h
 = 0.03
Y gd 
     
13.
 (a) [a] = [
 M 
1
T 
 – 
1
](b)
t k  ma
=
 t 
14.
6%
15.
 
q 
 =
 k  GM cr 
2
16.
 (b)
k 
 = 6.5 × 10
–3
17.
 (i) [
k 
] = [
 L
–2
](ii)
aa
si
µ
 -
l 
2
 (iii)
 I  I aa
sisi
µ µ
 -
224
l 
18.
a x ac
tan
-
 Ê Ë Á ˆ ¯ ˜  +
1
19.
 (0.054 0.003)s
20.
 (a) [a] = [
 M 
º
 L
1
T 
–2
]; [b] = [
 M 
1
 L
º
T 
–2
](b)
b
 = – 0.022
kg s
–2
21.
8.84 ± 0.24
ly
22.
 (a) 3.2 × 10
28
(b) 2.5 × 10
42
 (c) 1.5 × 10
6
23.
 5.6
m
SOLUTIONS
2.
 
q p 
= ∞ = ¥ = ¥=
36180363143618000628.....
rad 
radian
500
q 
= 3.6°
h 
10 mm
, find the value of h. Does the value surprise you. [For small
q 
take tan
qq q 
= +
3
3 and sec
q q 
= +
12
2
Also take
23174
23
..
=
]
Rh
4
 
L 
EVEL 
 1
Q. 1. A particle is travelling on a curved path. In an interval
t 
 its speed changed from
v
 to 2
v
. However, the change in magnitude of its velocity was found to be
∆
V v

=
. What can you say about the direction of velocity at the beginning and at the end of the interval (
 t 
)?Q. 2.
 
Two tourist
 A
 and
 B
 who are at a distance of 40
km
 from their camp must reach it together in the shortest possible time. They have one bicycle and they decide to use it in turn. ‘
 A
’ started walking at a speed of 5
km
 
hr 
–1
 and
 B
 moved on the bicycle at a speed of 15
km hr 
–1
. After moving certain distance
 B
 left the bicycle and walked the remaining distance.
 A
, on reaching near the bicycle, picks it up and covers the remaining distance riding it. Both reached the camp together. (a) Find the average speed of each tourist. (b) How long was the bicycle left unused?Q. 3.
 
The position time graph for a particle travelling along
 x 
 axis has been shown in the figure. State whether following statements are true of false. (a) Particle starts from rest at
t 
 = 0. 
t 
1
X O 
t 
2
 t 
3
t 
 (b) Particle is retarding in the interval 0 to
t 
1
 and accelerating in the interval
t 
1
 to
t 
2
. (c) The direction of acceleration has changed once during the interval 0 to
t 
3
Q. 4. The position time graph for a particle moving along
 X 
 axis has been shown in the fig. At which of the indicated points the particle has
KINEMATICS
 (i) negative velocity but acceleration in positive
 X 
 direction. (ii) positive velocity but acceleration in negative
 X 
 direction. (iii) received a sharp blow (a large force for negligible interval of time)?
X O AD 
B
 C E F G H t 
 
Q. 5. A particle is moving along positive
 X 
 direction and is retarding uniformly. The particle crosses the origin at time
t 
 = 0 and crosses the point
 x 
 = 4.0
m
 at
t 
 = 2
s
. (a) Find the maximum speed that the particle can possess at
 x 
 = 0. (b) Find the maximum value of retardation that the particle can have.
 
Q. 6. The velocity time graph for two particles (1 and 2) moving along
 X 
 axis is shown in fig. At time
t 
 = 0, both were at origin. (a) During first 4 second of motion what is maximum separation between the particles? At what time the separation is maximum? (b) Draw position (
 x 
) vs time (
t 
) graph for the particles for the given interval.
412O4
t s 
( )
v 
 (m/s)
02
4
 
 2.2
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
Q. 7. A ball travelling in positive
 X 
 direction with speed
V 
0
 hits a wall perpendicularly and rebounds with speed
V 
0
. During the short interaction time (
 t 
) the force applied by the wall on the ball varies as shown in figure.
t t 
O
F 
x 
 Draw the velocity-time graph for the ball during the interval 0 to
t 
 Q. 8. For a particle moving along a straight line consider following graphs
 A
,
 B
,
C 
 and
 D
. Here
 x 
,
v
 and
t 
 are position, velocity and time respectively. (i) In which of the graphs the magnitude of acceleration is decreasing with time? (ii) In which of the graphs the magnitude of acceleration is increasing with time? (iii) If the body is definitely going away from the starting point with time, which of the given graphs represent this condition.
t 
XOparabola
t 
O
t 
O
 t 
O
v v v 
(a)(c)(b)(d)
 
Q. 9. Two particles
 A
 and
 B
 start from same point and move along a straight line. Velocity-time graph for both of them has been shown in the fig. Find the maximum separation between the particles in the interval 0 <
t 
 < 5 sec.
O10
v 
 (m/s)A2 4B
t s 
( )
 
Q. 10. A particle starts from rest (at
 x 
 = 0) when an acceleration is applied to it. The acceleration of the particle changes with its co-ordinate as shown in the fig. Find the speed of the particle at
 x 
 = 10
m
.
O8
a 
x 
(m/s )
2
8X (in m)10
 Q. 11. Acceleration vs time graph for a particle moving along a straight line is as shown. If the initial velocity of the particle is
u
 = 10
m/s
, draw a plot of its velocity vs time for 0 <
t 
 < 8.
a
(m/s )1048
t 
(in s)
 Q. 12. The velocity (
V 
) – time (
t 
) graphs for two particles
 A
 and
 B
 moving rectilinearly have been shown in the figure for an interval of 2 second. (a) At
t 
 = 1
s
, which of the two particles (
 A
 or
 B
) has received a severe blow? (b) Draw displacement (
 X 
) – time (
t 
) graph for both of them.
o–41 24V (m/s)
t 
(s)(a)o–41 24V (m/s)
t 
(s)(b)
4
 
 K
INEMATICS
 
2.3
 
Q. 13. A particle starts moving rectilinearly at time
t 
 = 0 such that its velocity(
v
) changes with time (
t 
) as per equation – 
v
 = (
t 
2
 – 2
t 
)
m/s
 for 0
<
 
t 
 < 2
s
 = (–
t 
2
+ 6
t 
 – 8)
m/s
 for 2 <
t 
 < 4
s
 (a) Find the interval of time between
t 
 = 0 and
t 
 = 4
s
 when particle is retarding. (b) Find the maximum speed of the particle in the interval 0 <
t 
 < 4
s
.
 
Q. 14. Our universe is always expanding. The rate at which galaxies are receding from each other is given by Hubble’s law (discovered in 1929 by
 E 
. Hubble). The law states that the rate of separation of two galaxies is directly proportional to their separation. It means relative speed of separation of two galaxies, presently at separation
r 
 is given by
v
 =
 Hr 
 
 H 
 is a constant known as Hubble’s parameter. Currently accepted value of
 H 
 is 2.32 × 10
–18
 
s
–1
 (a) Express the value of
 H 
 in unit of  
 Km s
.
 −
1
Mega light year 
 (b) Find time required for separation between two galaxies to change from
r 
 to 2
r 
.Q. 15. A stone is projected vertically up from a point on the ground, with a speed of 20
m/s
. Plot the variation of followings with time during the entire course of flight – (a) Velocity (b) Speed (c) Height above the ground (d) distance travelled
 
Q. 16. A ball is dropped from a height
 H 
 above the ground. It hits the ground and bounces up vertically to a height where it is caught. Taking origin at the point from where the ball was dropped, plot the variation of its displacement
vs
 velocity. Take vertically downward direction as positive.
 
Q. 17. A helicopter is rising vertically up with a velocity of 5
ms
–1
. A ball is projected vertically up from the helicopter with a velocity
V 
 (relative to the ground). The ball crosses the helicopter 3 second after its projection. Find
V 
.
 
Q. 18. A chain of length
 L
 supported at the upper end is hanging vertically. It is released. Determine the interval of time it takes the chain to pass a point 2
 L
 below the point of support, if all of the chain is a freely falling body.
 
Q. 19. Two nearly identical balls are released simultaneously from the top of a tower. One of the balls fall with a constant acceleration of
g
1
 = 9.80
ms
–2
 while the other falls with a constant acceleration that is 0.1% greater than
g
1
. [This difference may be attributed to variety of reasons. You may point out few of them]. What is the displacement of the first ball by the time the second one has fallen 1.0
 mm
 farther than the first ball? Q. 20. Two projectiles are projected from same point on the ground in
 x 
-
 y
 plane with
 y
 direction as vertical. The initial velocity of projectiles are 
V V i V jV V i V j
 x y x y
11222
1
= += +
ˆˆˆˆ It is given that
V 
 x 
1
>
V 
 x 
2
 and
V 
 y
1
<
V 
 y
2
.
 
Check whether all of the following statement/s are True. (a) Time of flight of the second projectile is greater than that of the other. (b) Range of first projectile may be equal to the range of the second. (c) Range of the two projectiles are equal if
V 
 x 
1
V 
 y
1
=
V 
 x 
2
V 
 y
2
 (d) The projectile having greater time of flight can have smaller range. Q. 21.
 
(a)
 
A particle starts moving at
t 
 = 0 in
 x 
-
 y
 plane such that its coordinates (in cm) with time (in sec) change as
 x 
 = 3
t 
 and
 y
 = 4 sin (3
t 
). Draw the path of the particle.
 
(b)
 
If position vector of a particle is given by
r t t i t t j
= -
( )
 + -
( )
416312
22
ˆˆ
, then find distance travelled in first 4 sec.
 
Q. 22. Two particles projected at angles
q 
1
 and
q 
2
 (<
q 
1
) to the horizontal attain same maximum height. Which of the two particles has larger range? Find the ratio of their range. Q. 23. A ball is projected from the floor of a long hall having a roof height of
 H 
 = 10
m
. The ball is projected with a velocity of
u
 = 25
ms
–1
 making an angle of
q 
 = 37° to the horizontal. On hitting the roof the ball loses its entire vertical component of velocity but there is no change in the horizontal component of its velocity. The ball was projected
4
 
 2.4
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
from point
 A
 and it hits the floor at
 B
. Find distance
 AB
. 
HBA
u 
 
 Q. 24. In a tennis match Maria Sharapova returns an incoming ball at an angle that is 4° below the horizontal at a speed of 15
m/s
. The ball was hit at a height of 1.6
m
above the ground. The opponent, Sania Mirza, reacts 0.2
s
 after the ball is hit and runs to the ball and manages to return it  just before it hits the ground. Sania runs at a speed of 7.5
m/s
 and she had to reach 0.8
m
 forward, from where she stands, to hit the ball. (a) At what distance Sania was standing from Maria at the time the ball was returned by Maria? Assume that Maria returned the ball directly towards Sania. (b) With what speed did the ball hit the racket of Sania? [
g
 = 9.8
m/s
2
] Q.25. A player initially at rest throws a ball with an initial speed
u
 = 19.5
m/s
 at an angle
θ 
 =     
−
sin
 1
 1213
 to the horizontal. Immediately after throwing the ball he starts running to catch it. He runs with constant acceleration (
a
) for first 2
s
 and thereafter runs with constant velocity. He  just manages to catch the ball at exactly the same height at which he threw the ball. Find ‘
a
’. Take
g
 = 10
 m/s
2
. Do you think anybody can run at a speed at which the player ran?Q. 26. In a cricket match, a batsman hits the ball in air. A fielder, originally standing at a distance of 12
m
 due east of the batsman, starts running 0.6 s after the ball is hits. He runs towards north at a constant speed of 5
m/s
 and just manages to catch the ball 2.4
s
 after he starts running. Assume that the ball was hit and caught at the same height and take
g
 = 10
m/s
2
g
 = 10
m/s
2
 Find the speed at which the ball left the bat and the angle that its velocity made with the vertical.
 
Q. 27. The time of flight, for a projectile, along two different paths to get a given range
 R
, are in ratio 2 : 1. Find the ratio of this range
 R
 to the maximum possible range for the projectile assuming the projection speed to be same in all cases. Q. 28. A boy ‘
 A
’ is running on a circular track of radius
 R
. His friend, standing at a point O on the circumference of the track is throwing balls at speed
u
 =
gR
. Balls are being thrown randomly in all possible directions. Find the length of the circumference of the circle on which the boy is completely safe from being hit by a ball.
CORAo
 Q. 29. A rectangular cardboard
 ABCD
has dimensions of 40
cm
 × 30
cm
. It is moving in a direction perpendicular to its shorter side at a constant speed of 2
cm
 / 
s
. A small insect starts at corner
 A
 and moves to diagonally opposite corner
C 
. On reaching
C 
 it immediately turns back and moves to
 A
. Throughout the motion the insect maintains a constant speed relative to the board. It takes 10
s
 for the insect to reach
C 
 starting from
 A
. Find displacement and distance travelled by the insect in reference frame attached to the ground in the interval the insect starts from
 A
 and comes back to
 A
.
DCBA40 cm30 cm2 cm/s
 Q. 30. Two particles
 A
 and
 B
 separated by 10
m
 at time
t
= 0 are moving uniformly.
 A
 is moving along line
 AB
 at a constant velocity of 4
m/s
 and
 B
 is moving perpendicular to the velocity of
 A
 at a constant velocity of 5
m/s
. After what time the two particles will be nearest to each other?
AB4 m/s10 m5 m/s
4
 
 K
INEMATICS
 
2.5
 Q. 31. Four cars are moving along a straight road in the same direction. Velocity of car 1 is 10
m/s
. It was found that distance between car 1 and 2 is decreasing at a rate of 2
m/s
, whereas driver in car 4 observed that he was nearing car 2 at a speed of 8
m/s
. The gap between car 2 and 3 is decreasing at a rate of 3
m/s
.
20 m20 m20 m213410 m/s
 (a) If cars were at equal separations of 20
m
 at time
t 
 = 0, after how much time
t 
0
 will the driver of car 2 see for the first time, that another car overtakes him? (b) Which car will be first to overtake car 1? Q. 32. Acceleration of a particle as seen from two reference frames 1 and 2 has magnitude 3
m/s
2
 and 4
m/s
2
 respectively. What can be magnitude of acceleration of frame 2 with respect to frame 1? Q. 33. A physics professor was driving a Maruti car which has its rear wind screen inclined at
q 
 = 37° to the horizontal. Suddenly it started raining with rain drops falling vertically. After some time the rain stopped and the professor found that the rear wind shield was absolutely dry. He knew that, during the period it was raining, his car was moving at a constant speed of
V 
c
 = 20
km
 / 
hr 
. [tan 37° = 0.75] (a) The professor calculated the maximum speed of vertically falling raindrops as
V 
max
. What is value of
V 
max
 that he obtained. (b) Plot the minimum driving speed of the car vs. angle of rear wind screen with horizontal (
q 
) so as to keep rain off the rear glass. Assume that rain drops fall at constant speed
V 
r 
 Q. 34.
 TA BCC
 A train(
T 
) is running uniformly on a straight track. A car is travelling with constant speed along section AB of the road which is parallel to the rails. The driver of the car notices that the train is having a speed of 7
m/s
 with respect to him. The car maintains the speed but takes a right turn at
 B
 and travels along
 BC 
. Now the driver of the car finds that the speed of train relative of him is 13
m/s
. Find the possible speeds of the car.
V 
2
B A
V 
1
 A police car
 B
 is chasing a culprit’s car
 A
. Car
 A
 and
 B
 are moving at constant speed
V 
1
 = 108
km
 / 
hr 
 and
V 
2
 = 90
km
 / 
hr 
 respectively along a straight line. The police decides to open fire and a policeman starts firing with his machine gun directly aiming at car A. The bullets have a velocity
u
 = 305
m/s
 relative to the gun. The policeman keeps firing for an interval of
T 
0
 = 20
s
. The Culprit experiences that the time gap between the first and the last bullet hitting his car is
D
t 
. Find
D
t 
. Q. 36. A chain of length
 L
 is supported at one end and is hanging vertically when it is released. All of the chain falls freely with acceleration
g
. The moment, the chain is released a ball is projected up with speed u from a point 2
 L
 below the point of support. Find the interval of time in which the ball will cross through the entire chain. Q. 37.Jet plane
 A
 is moving towards east at a speed of 900
km
 / 
hr 
. Another plane
 B
 has its nose pointed towards 45°
 N 
 of
 E 
 but appears to be moving in direction 60°
 N 
 of
W 
 to the pilot in A. Find the true velocity of
 B
. [sin 60° = 0.866 ; sin 75° = 0.966]
45°E60°BN
AE
Q. 38. A small cart A starts moving on a horizontal surface, assumed to be
 x 
-
 y
 plane along a straight line parallel to
 x 
-axis (see figure) with a constant acceleration of 4
m
 / 
s
2
. Initially it is located on the positive
 y
-axis at a distance 9
m
 from origin. At
5
 
 2.6
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
the instant the cart starts moving, a ball is rolled along the surface from the origin in a direction making an angle 45° with the
 x 
-axis. The ball moves without friction at a constant velocity and hits the cart. (a)
 
Describe the path of the ball in a reference frame attached to the cart. (b) Find the speed of the ball.Q. 39. (a)A boy on a skateboard is sliding down on a smooth incline having inclination angle
q 
. He throws a ball such that he catches it back after time
T 
. With what velocity was the ball thrown by the boy relative to himself ?
q 
 (b) Barrel of an anti aircraft gun is rotating in vertical plane (it is rotating up from the horizontal position towards vertical orientation in the plane of the fig). The length of the barrel is
 L
= 2 
m
 and barrel is rotating with angular velocity
w 
 = 2
rad 
 / 
s
. At the instant angle
q 
 is 45° a shell is fired with a velocity 22
m/s
 with respect to the exit point of the barrel. The tank recoils with speed 4
m/s
. What is the launch speed of the shell as seen from the ground?
q 
Q. 40
.
 
long piece of paper is10
cm
 wide and is moving uniformly along its length with a velocity of 2
cm
 / 
s
. An ant starts moving on the paper from point
 A
 and moves uniformly with respect to the paper. A spider was located exactly opposite to the ant just outside the paper at point
 B
 at the instant the ant started to move on the paper. The spider, without moving itself, was able to grab the ant 5 second after it (the ant) started to move. Find the speed of ant relative to the paper.
 AO X Y 
45°
A
B
2 cm/s10 cm
Q.
 
41. Two particles A and B are moving uniformly in a plane in two concentric circles. The time period of rotation is T
A
 = 8 minute and T
B
 = 11 minute respectively for the two particles. At time
t 
 = 0, the two particles are on a straight line passing through the centre of the circles. The particles are rotating in same sense. Find the minimum time when the two particles will again fall on a straight line passing through the centre.Q.
 
42. A particle moves in
 xy
 plane with its position vector changing with time (
t 
) as 
r t i t j
=sin +
( ) ( )
ˆcosˆ
 (in meter) Find the tangential acceleration of the particle as a function of time. Describe the path of the particle.Q.
 
43. Two paper discs are mounted on a rotating vertical shaft. The shaft rotates with a constant angular speed
w 
 and the separation between the discs is H. A bullet is fired vertically up so that it pierces through the two discs. It creates holes H1 and H2 in the lower and the upper discs. The angular separation between the two holes (measured with respect to the shaft axis) is
q 
. Find the speed (
v
) of the bullet. Assume that the speed of the bullet does not change while travelling through distance H and that the discs do not complete even one revolution in the interval the bullet pierces through them.
HH1H2
V
Q. 44. (a) A car moves around a circular arc subtending an angle of 60° at the centre. The car moves at a constant speed
u
0
 and magnitude of its
4
 
 K
INEMATICS
 
2.7
instantaneous acceleration is
a
0
. Find the average acceleration of the car over the 60° arc.(b)
 
The speed of an object undergoing uniform circular motion is 4
m/s
. The magnitude of the change in the velocity during 0.5 sec is also 4
m/s
. Find the minimum possible centripetal acceleration (in
m/s
2
) of the object. Q. 45.
 
A particle is fixed to the edge of a disk that is rotating uniformly in anticlockwise direction about its central axis. At time
t 
 = 0 the particle is on the
 X 
 axis at the position shown in figure and it has velocity
v
y x v t 
at = 0
 (a) Draw a graph representing the variation of the
 x 
 component of the velocity of the particle as a function of time. (b) Draw the
 y
-component of the acceleration of the particle as a function of time.Q. 46. A disc is rotating with constant angular velocity
w 
 in anticlockwise direction. An insect sitting at the centre (which is origin of our co-ordinate system) begins to crawl along a radius at time
t 
 = 0 with a constant speed
V 
 relative to the disc. At time
t 
 = 0 the velocity of the insect is along the
 X 
 direction. (a) Write the position vector
( )
 of the insect at time ‘
t 
’. (b) Write the velocity vector
( )
 of the insect at time ‘
t 
’. (c) Show that the
 X 
 component of the velocity of the insect become zero when the disc has rotated through an angle
q 
 given by tan
q q 
=
1. 
O
v 
w 
X
y 
 
Q. 47. (a) A point moving in a circle of radius
 R
 has a tangential component of acceleration that is always
n
 times the normal component of acceleration (radial acceleration). At a certain instant speed of particle is
v
0
. What is its speed after completing one revolution?(b)The tangential acceleration of a particle moving in
 xy
 plane is given by
a
t 
 =
a
0
 cos
q 
. Where
a
0
 is a positive constant and
q 
 is the angle that the velocity vector makes with the positive direction of
 X 
 axis. Assuming the speed of the particle to be zero at
 x 
 = 0, find the dependence of its speed on its
 x 
 co-ordinate.Q. 48. A particle is rotating in a circle. When it is at point
 A
 its speed is
V 
. The speed increases to 2
V
by the time the particle moves to
 B
. Find the magnitude of change in velocity of the particle as it travels from
 A
 to
 B
. Also, find
V V 
 A

D
; where
V 
 A
 is its velocity at point A and
D
V 

 is change in velocity as it moves from
 A
 to
 B
.Q. 49. A particle starts from rest moves on a circle with its speed increasing at a constant rate of . Find the angle through which it 0.8
 ms
–2
 would have turned by the time its acceleration becomes 1
ms
2
.Q. 50. In the arrangement shown in the fig, end
 A
 of the string is being pulled with a constant horizontal velocity of 6
m/s
. The block is free to slide on the horizontal surface and all string segments are horizontal. Find the velocity of point
P
 on the thread.
A6 m/sP
 Q. 51. In the arrangement shown in the fig, block
 A
 is pulled so that it moves horizontally along the line
 AX 
 with constant velocity
u
. Block
 B
 moves along the incline. Find the time taken by
 B
 to reach the pulley
P
 if
u
 = 1
m/s
. The string is inextensible.
2 mA
u 
PB
q 
=30
0
 1 2  m
X
5
 
 2.8
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
L 
EVEL 
 2
Q. 52. Two friends
 A
 and
 B
 are running on a circular track of perimeter equal to 40
m
. At time
t 
 = 0 they are at same location running in the same direction.
 A
 is running slowly at a uniform speed of 4.5
km
 / 
hr 
 whereas
 B
 is running swiftly at a speed of 18
km
 / 
hr 
. (a) At what time
t 
0
 the two friends will meet again? (b) What is average velocity of
 A
 and
 B
 for the interval
t 
 = 0 to
t 
 =
t 
0
?Q. 53. A particle is moving along
 x 
 axis. Its position as a function of time is given by
 x 
 =
 x 
(
t 
). Say whether following statements are true or false. (a) The particle is definitely slowing down if  
d xdt dxdt 
22
 0 0
> <
and
 (b) The particle is definitely moving towards the origin if
d xdt 
2
0
( )
<
Q. 54. Graph of position (
 x 
) vs inverse of velocity
    
 for a particle moving on a straight line is as shown. Find the time taken by the particle to move from
 x 
 = 3
m
 to
 x 
 = 15
m
. 
x m 
()153281
v 
(sm )
–1
O
Q. 55. Harshit and Akanksha both can run at speed
v
 and walk at speed
u
(
u
 <
v
). They together start on a journey to a place that is at a distance equal to
 L
. Akanksha walks half of the distance and runs the second half. Harshit walks for half of his travel time and runs in the other half. (a) Who wins? (b) Draw a graph showing the positions of both Harshit and Akanksha versus time. (c) Find Akanksha’s average speed for covering distance
 L
. (d) How long does it take Harshit to cover the distance?Q. 56. There are two cars on a straight road, marked as
 x 
 axis. Car
 A
 is travelling at a constant speed of
V 
 A
 = 9
m/s.
.
 
Let the position of the Car
 A
, at time
t 
 = 0, be the origin. Another car
 B
 is
 L
 = 40
m
ahead of car
 A
 at
t 
 = 0 and starts moving at a constant acceleration of
a
 = 1
m/s
2
 (at
t 
 = 0). Consider the length of the two cars to be negligible and treat them as point objects.
A= 9 m/s
$
V 
A
B= 1m/s
$
a 
2
x 
= 0
x 
L = 40 m
 (a) Plot the position–time (
 x 
–
t 
) graph for the two cars on the same graph. The two graphs intersect at two points. Draw conclusion from this. (b) Determine the maximum lead that car
 A
 can have.Q. 57. Particle
 A
 is moving with a constant velocity of
V 
 A
 = 50
ms
–1
 in positive
 x 
 direction. It crossed the origin at time
t
= 10
s
. Another particle
 B
 started at
t 
 = 0 from the origin and moved with a uniform acceleration of
a
 B
 = 2
ms
–2
 in positive
 x 
 direction. (a) For how long was
 A
 ahead of
 B
 during the
subsequent journey? (b) Draw the position (
 x 
) time (
t 
) graph for the two particles and mark the interval for which
 A
 was ahead of
 B
. Q. 58. (a) A particle is moving along the
 x 
 axis and its velocity vs position graph is as shown. Is the acceleration of the particle increasing, decreasing or remains constant? 
x 
 (b) A particle is moving along
 x 
 axis and its velocity (
v
) vs position (
 x 
) graph is a curve as shown in the figure. Line
 APB
 is normal to the curve at point
P
. Find the instantaneous acceleration of the particle at
 x 
 = 3.0
m
.
4
 
 K
INEMATICS
 
2.9
O C(3,0)B(4,0)
x m 
(in )
P Av m/s 
( )
 
Q. 59. A particle has co-ordinates (
 x 
,
 y
). Its position vector makes on angle
q 
 with positive
 x 
 direction. In an infinitesimally small interval of time the particle moves such that length of its position vector does not change but angle
q 
 increases by d
q 
. Express the change in position vector of the particle in terms of
 x 
,
 y
,
d 
q 
 and unit vectors ˆ
i
 and ˆ
 j
.
x r 
q 
P 
(x,y) 
O
 Q. 60. A rope is lying on a table with one of its end at point
O
 on the table. This end of the rope is pulled to the right with a constant acceleration starting
from rest. It was observed that last 2
m
 length of the rope took 5
s
 in crossing the point
O
 and the last 1
m
 took 2
s
 in crossing the point
O
.
O
 (a) Find the time required by the complete rope to travel past point
O
. (b) Find length of the rope. Q. 61.
x x 
0 
 u t 
0
 t 
0
O  O t  t 
 
Two particles 1 and 2 move along the
 x 
 axis. The position (
 x 
) - time (
t 
) graph for particle 1 and velocity (
v
) - time (
t 
) graph for particle 2 has been shown in the figure. Find the time when the two particles collide. Also find the position (
 x 
) where they collide. It is given that
 x 
0
 =
ut 
0
, and that the particle 2 was at origin at
t 
 = 0.
 
Q. 62. Two stations
 A
 and
 B
 are 100
km
 apart. A passenger train crosses station
 A
 travelling at a speed of 50
km
 / 
hr 
. The train maintains constant speed for 1 hour 48 minute and then the driven applies brakes to stop the train at station
 B
 in next 6 minute. Another express train starts from station
 B
 at the time the passenger train was crossing station
 A
. The driver of the express train runs the train with uniform acceleration to attain a peak speed
v
0
. Immediately after the train attains the peak speed
v
0
, he applies breaks which cause the train to stop at station
 A
 at the same time the passenger train stops at
 B
. Brakes in both the trains cause uniform retardation of same magnitude. Find the travel time of two trains and
v
0
.
 
Q. 63. Particle
 A
 starts from rest and moves along a straight line. Acceleration of the particle varies with time as shown in the graph. In 10
s
 the velocity of the particle becomes 60
m/s
 and the acceleration drops to zero. Another particle
 B
 starts from the same location at time
t 
 = 1.1
s
 and has acceleration – time relationship identical to
 A
 with a delay of 1.1
s
. Find distance between the particles at time
t 
 = 15
s
.
a
BAO1.1 10
t s 
( )
Q. 64.
a
x 
a
y 
t 
a
0
 a
0
t t 
0
 t 
0
O O2
 0
t 
 2
t 
0
3
 0
t 
 3
t 
0
4
t 
0
 4
t 
0
 A particle is moving in
 x 
–
 y
 plane. The
 x 
 and
 y
 components of its acceleration change with time according to the graphs given in figure. At time
t 
 = 0, its velocity is
v
0
 directed along positive
5
 
 2.10
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
 y
direction. If
a vt 
000
, find the angle that the velocity of the particle makes with
 x 
 axis at time
t 
 = 4
t 
0
. Q. 65. A particle is moving along positive
 x 
 direction and experiences a constant acceleration of 4
m/s
2
 in negative
 x 
 direction. At time
t 
 = 3 second its velocity was observed to be 10
m/s
 in positive
 x 
 direction. (a) Find the distance travelled by the particle in the interval
t 
 = 0 to
t 
 = 3
s
. Also find distance travelled in the interval
t 
 = 0 to
t 
 = 7.5
s.
. (b) Plot the displacement – time graph for the interval
t 
 = 0 to 7.5
s
.Q. 66. A bead moves along a straight horizontal wire of length
 L
, starting from the left end with velocity
v
0
. Its retardation is proportional to the distance that remains to the right end of the wire. Find the initial retardation (at left end of the wire) if the bead reaches the right end of the wire with a velocity
v
0
2.Q. 67. A ball is projected vertically up from the ground surface with an initial velocity of
u
 = 20
m/s
.
O
 is a fixed point on the line of motion of the ball at a height of
 H 
 = 15
m
 from the ground. Plot a graph showing variation of distance (
s
) of the ball from the fixed point
O,
 with time (
t 
). [Take
g
 = 10
m/s
2
]. Plot the graph for the entire time of flight of the ball.
H u O 
 
Q. 68. Two bodies 1 and 2 of different shapes are released on the surface of a deep pond. The mass of the two bodies are
m
1
 = 1
kg
 and
m
2
 = 1.2
kg
 respectively. While moving through water, the bodies experience resistive force given as
 R
 =
bv
, where
v
 is speed of the body and
b
 is a positive constant dependent on shape of the body. For bodies 1 and 2 value of
b
 is 2.5
kg
 / 
s
 and 3.0
kg
 / 
s
 respectively. Neglect all other forces apart from gravity and the resistive force, while answering following questions : [Hint : acceleration = force/mass] (i) With what speed
v
10
 and
v
20
 will the two bodies hit the bed of the pond. [Take
g
 = 10
m
 / 
s
2
] (ii) Which body will acquire speed equal to half the terminal speed in less time. Q. 69. A prototype of a rocket is fired from the ground. The rocket rises vertically up with a uniform acceleration of 54 
m/s
2
. 8 second after the start a small nut gets detached from the rocket. Assume that the rocket keeps rising with the constant acceleration. (a) What is the height of the rocket at the instant the nut lands on the ground (b) Plot the velocity – time graph for the motion of the nut after it separates from the rocket till it hits the ground. Plot the same velocity– time graph in the reference frame of the rocket. Take vertically upward direction as positive and
g
 = 10
m/s
2
 Q. 70. An elevator starts moving upward with constant acceleration. The position time graph for the floor of the elevator is as shown in the figure. The ceiling to floor distance of the elevator is 1.5
m
. At
t 
 = 2.0
s
, a bolt breaks loose and drops from the ceiling. (a) At what time
t 
0
 does the bolt hit the floor? (b) Draw the position time graph for the bolt starting from time
t 
 = 0. [take g = 10
m
 / 
s
2
]
(In second)(In meter)
y 
4.0O2.0
t 
 
Q. 71. At
t 
 = 0 a projectile is projected vertically up with a speed
u
 from the surface of a peculiar planet. The acceleration due to gravity on the planet changes linearly with time as per equation
g
 =
a 
t 
 where
a 
 is a constant.
5
 
 K
INEMATICS
 
2.11
 (a) Find the time required by the projectile to attain maximum height. (b) Find maximum height attained. (c) Find the total time of flight.
 
Q. 72. A wet ball is projected horizontally at a speed of
u
 = 10
m
 / 
s
 from the top of a tower
h
 = 31.25
m
 high. Water drops detach from the ball at regular intervals of
D
t 
 = 1.0
s
 after the throw. (a) How many drops will detach from the ball before it hits the ground. (b) How far away the drops strike the ground from the point where the ball hits the ground?
 
Q. 73. Two stones of mass
m
 and
 M 
 (
 M 
 >
m
) are dropped
D
t 
 time apart from the top of a tower. Take time
t 
 = 0 at the instant the second stone is released. Let
D
v
and
D
s
 be the difference in their speed and their mutual separation respectively. Plot the variation of
D
v
 and
D
s
 with time for the interval both the stones are in flight. [
g
 = 10
m
 / 
s
2
]
 
Q. 74. A particle is moving in the
 xy
 plane on a sinusoidal course determined by
 y
 =
 A
 sin
kx 
, where
k 
 and
 A
 are constants. The
 X 
 component of the velocity of the particle is constant and is equal to
v
0
 and the particle was at origin at time
t 
 = 0. Find the magnitude of the acceleration of the particle when it is at point having
 x 
 co ordinate
 x 
 =
p 
2
k 
.
 
Q. 75. A ball is projected from a cliff of height
h
 = 19.2
m
 at an angle
a 
 to the horizontal. It hits an incline passing through the foot of the cliff, inclined at an angle
q 
 to the horizontal. Time of flight of the ball is
T 
 = 2.4
s
. Foot of the cliff is the origin of the co-ordinate system, horizontal is
 x 
 direction and vertical is
 y
 direction (see figure). Plot of
 y
 co-ordinate
vs
 time and
 y
 component of velocity of the ball (
v
 y
)
vs
 its
 x 
 co-ordinate (
 x 
) is as shown.
 x 
 and
 y
 are in
m
 and time is in
s
 in the graph. [g = 10
m
 / 
s
2
]
A
y x O h 
q a 
y(m) v 
y 
v 
0 
–v 
0 
x 
38.4
t(s) O 
 2.419.2
 (a) Find the angle of projection
a 
 (b) Find the inclination (
q 
) of the incline. (c) If the ball is projected with same speed but at an angle
q 
 (= inclination of incline) to the horizontal, will it hit the incline above or below the point where it struck the incline earlier? Q. 76. (i) A canon can fire shells at speed
u
. Inclination of its barrel to the horizontal can be changed in steps of
q
= 1° ranging from
q 
1
 = 15° to
q 
2
 
= 85°. Let
 R
n
 be the horizontal range for projection angle
q
=
n
°. 
∆
 R R R
n n n
= −
+
1
 For what value of
n
 the value of
 R
n
 is maximum? Neglect air resistance. (ii) A small water sprinkler is in the shape of a hemisphere with large number of uniformly spread holes on its surface. It is placed on ground and water comes out of each hole with speed u. Assume that we mentally divide the ground into many small identical patches – each having area
S 
. What is the distance of a patch from the sprinkler which receives maximum amount of water ? A gun fires a large number of bullets upward. Due to shaking of hands some bullets deviate as much as 1° from the vertical. The muzzle speed of the gun is 150
m/s
 and the height of gun above the ground is negligible. The radius of the head of the person firing the gun is 10
cm
. You can assume that acceleration due to gravity is nearly constant for heights involved and its value is g = 10
m
 / 
s
2
. The gun fires 1000 bullets and they fall uniformly over a circle of radius
r 
. Neglect air resistance. You can use the fact sin
q 
 ~
q 
 when
q 
 is small. (a) Find the approximate value of
r 
. (b) What is the probability that a bullet will fall on the person’s head who is firing? Three stones are projected simultaneously with same speed
u
 from the top of a tower. Stone 1 is
5
 
 2.12
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
projected horizontally and stone 2 and stone 3 are projected making an angle
q 
 with the horizontal as shown in fig. Before stone 3 hits the ground, the distance between 1 and 2 was found to increase at a constant rate
u
.
321
q q 
 (a) Find
q 
 (b) Find the rate at which the distance between 2 and 3 increases. A horizontal electric wire is stretched at a height
h
 = 10
m
 above the ground. A boy standing on the ground can throw a stone at a speed
u
 = 20
ms
–1
. Find the maximum horizontal distance
 x 
 at which a bird sitting on the wire can be hit by the stone.
h x u 
q 
Q. 80. A wall
OP
 is inclined to the horizontal ground at an angle
a 
. Two particles are projected from points
 A
 and
 B
 on the ground with same speed (
u
) in directions making an angle
q 
 to the horizontal (see figure). Distance between points
 A
 and
 B
 is
 x 
0
 = 24
m
. Both particles hit the wall elastically and fall back on the ground. Time of flight (time required to hit the wall and then fall back on to the ground) for particles projected from
 A
 and
 B
 are 4
s
 and 2
s
 respectively. Both the particles strike the wall perpendicularly and at the same location. [In elastic collision, the velocity component of the particle that is perpendicular to the wall gets reversed without change in magnitude] (a) Calculate maximum height attained by the particle projected from
 A
.
APB
q a q 
u 
O
u x 
0
 (b) Calculate the inclination of the wall to the horizontal (
a 
) [
g
 = 10
m
 / 
s
2
] Q. 81.
 AB
 is a pipe fixed to the ground at an inclination of 37°. A ball is projected from point
O
 at a speed of
u
 = 20
m/s
 at an angle of 53° to the horizontal and it smoothly enters into the pipe with its velocity parallel to the axis of the pipe. [Take
g
 = 10
ms
–2
]
u O B A
37°53°
L   
 (a) Find the length
 L
 of the pipe (b) Find the distance of end
 B
 of the pipe from point
O
.
 
Q. 82. (a) A boy throws several balls out of the window of his house at different angles to the horizontal. All balls are thrown at speed
u
 = 10
m
 / 
s
 and it was found that all of them hit the ground making an angle of 45° or larger than that with the horizontal. Find the height of the window above the ground [take
g
 = 10
m
 / 
s
2
]
 
(b)
 
A gun is mounted on an elevated platform
 AB
. The distance of the gun at A from the edge
 B
 is
 AB
 = 960
m
. Height of platform is
OB
 = 960
m
. The gun can fire shells with a velocity of
u
 = 100
m
 / 
s
 at any angle. What is the minimum distance (
OP
) from the foot of the platform where the shell of gun can reach?
u 
ABOP
 Q. 83
 
An object A is kept fixed at the point
 x 
 = 3
m
 
4
 
 K
INEMATICS
 
2.13
and
 y
 = 1.25
m
 on a plank
P
 raised above the ground. At time
t 
 = 0 the plank starts moving along the +
 x 
 direction with an acceleration 1.5
m
 / 
s
2
. At the same instant a stone is projected from the origin with a velocity
u
 as shown. A stationary person on the ground observes the stone hitting the object during its downwards motion at an angle of 45º to the horizontal. All the motions are in
 x 
-
 y
 plane. Find
u
 and the time after which the stone hits the object. Take
g
 = 10
m
 / 
s
2
1.25 m
y  AP x u 
O3 m
 Q. 84. (a) A particle is thrown from a height
h
horizontally towards a vertical wall with a speed
v
 as shown in the figure. If the particle returns to the point of projection after suffering two elastic collisions, one with the wall and another with the ground, find the total time of flight. [Elastic collision means the velocity component perpendicular to the surface gets reversed during collision.]
h v 
 (b) Touching a hemispherical dome of radius
 R
 there is a vertical tower of height
 H 
 = 4
 R
. A boy projects a ball horizontally at speed
u
 from the top of the tower. The ball strikes the dome at a height
 R
2 from ground and rebounds. After rebounding the ball retraces back its path into the hands of the boy. Find
u
.
HRO
 Q. 85. A city bus has a horizontal rectangular roof and a rectangular vertical windscreen. One day it was raining steadily and there was no wind. (a) Will the quantity of water falling on the roof in unit time be different for the two cases (i) the bus is still (ii) the bus is moving with speed
v
 on a horizontal road ? (b) Draw a graph showing the variation of quantity of water striking the windscreen in unit time with speed of the bus (
v
). Q. 86. A truck is travelling due north descending a hill of slope angle
q 
 = tan
–1
 (0.1) at a constant speed of 90
km
 / 
hr 
. At the base of the hill there is a gentle curve and beyond that the road is level and heads 30° east of north. A south bound police car is travelling at 80
km
 / 
hr 
 along the level road at the base of the hill approaching the truck. Find the velocity of the truck relative to police car in terms of unit vectors ˆ,ˆ
i j
 and ˆ
k 
. Take
 x 
 axis towards east,
 y
 axis towards north and
 z
 axis vertically upwards. Q. 87. Two persons
 A
 and
 B
 travelling at 60
km
 / 
hr 
–1
 in their cars moving in opposite directions on a straight road observe an airplane. To the person
 A
, the airplane appears to be moving perpendicular to the road while to the observe
 B
 the plane appears to cross the road making an angle of 45°. (a) At what angle does the plane actually cross the road (relative to the ground). (b) Find the speed of the plane relative to the ground. Q. 88.
C
l u 
A B
L
 Two friends
 A
 and
 B
 are standing on a river bank
 L
 distance apart. They have decided to meet at a point
C 
 on the other bank exactly opposite to
 B
. Both of them start rowing simultaneously on boats which can travel with velocity
V 
 = 5
km
 / 
hr 
 in still water. It was found that both reached at
C 
 at the same time. Assume that path of
4
 
 2.14
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
both the boats are straight lines. Width of the river is
l
 = 3.0
km
 and water is flowing at a uniform speed of
u
 = 3.0
km
 / 
hr 
. (a) In how much time the two friends crossed the river. (b) Find
 L
.
 Q. 89.
On a frictionless horizontal surface, assumed to be the
 x 
-
 y
 plane, a small trolley A is moving along a straight line parallel to the y-axis (see figure) with a constant velocity of (3 – 1)
m/s
. At a particular instant, when the line
OA
 makes an angle of 45° with the
 x 
-axis, a ball is thrown along the surface from the origin
O
. Its velocity makes an angle
f 
 with the
 x 
-axis and it hits the trolley.
y 
A
O 
45°
x 
 (a) The motion of the ball is observed from the frame of the trolley. Calculate the angle
q 
 made by the velocity vector of the ball with the
 x 
-axis in this frame. (b) Find the speed of the ball with respect to the surface, if
φ  θ 
=
43
 .
 Q. 90.
 A large heavy box is sliding without friction down a smooth plane having inclination angle
q 
. From a point
P
 at the bottom of a box, a particle is projected inside the box. The initial speed of the particle with respect to box is
u
 and the direction of projection makes an angle
a 
 with the bottom as shown in figure 
P Q
 (a) Find the distance along the bottom of the box between the point of projection
P
 and the point
Q
 where the particle lands. (Assume that the particle does not hit any other surface of the box. Neglect air resistance) (b) If the horizontal displacement of the particle as seen by an observer on the ground is zero, find the speed of the box with respect to the ground at the instant when the particle was projected.Q. 91. A ball is projected in vertical
 x 
–
 y
 plane from a car moving along horizontal
 x 
 direction. The car is speeding up with constant acceleration. Which one of the following trajectory of the ball is not possible in the reference frame attached to the car? Give reason for your answer. Explain the condition in which other trajectories are possible. Consider origin at the point of projection.
(a)(c)(b)(d)
y y y y x x x x 
Q. 92. A boy standing on a cliff 50
m
 high throws a ball with speed 40
m
 / 
s
 directly aiming towards a man standing on ground at
 B
. At the same time the man at
 B
 throws a stone with a speed of 10
m
 / 
s
 directly aiming towards the boy.
AB C 
 
50 m50 m
 (a) Will the ball and the stone collide? If yes, at what time after projection? (b) At what height above the ground the two objects collide? (c) Draw the path of ball in the reference frame of the stone.Q. 93. A man walking downhill with velocity
V 
0
 finds that his umbrella gives him maximum protection from rain when he holds it such that the stick is
5
 
 K
INEMATICS
 
2.15
perpendicular to the hill surface. When the man turns back and climbs the hill with velocity
V 
0
, he finds that it is most appropriate the hold the umbrella stick vertical. Find the actual speed of raindrops in terms of
V 
0
. The inclination of the hill is
q 
 = 37°.
q 
 
   V
  0
   V
  0
 Q. 94. There are two hills
 A
 and
 B
 and a car is travelling towards hill
 A
 along the line joining the two hills. Car is travelling at a constant speed
u
. There is a wind blowing at speed
u
 in the direction of motion of the car (i.e., from hill
 B
 to
 A
). When the car is at a distance
 x 
1
 from
 A
 and
 x 
2
 from
 B
 it sounds horn (for very short interval). Driver hears the echo of horn from both the hills at the same time.
Wind (u)
x 
2
 x 
1
u 
 AB
 Find the ratio
12
 taking speed of sound in still air to be
V 
. Q. 95. The figure shows a square train wagon
 ABCD
 which has a smooth floor and side length of 2
 L
. The train is mov-ing with uniform acceleration (
a
) in a direction parallel to
 DA
. A 'ball is rolled along the floor with a veloci-ty
u
, parallel to
 AB
, with respect to the wagon. The ball passes through the centre of the wagon floor. At the instant it is at the centre, brakes are
C DAB
u 
2
L
2
L
applied and the train begins to retard at a uniform rate that is equal to its previous acceleration (
a
)
 
(a) Will the ball hit the wall
 BC 
 or wall
CD
 or the corner
C 
?
 
(b) What is speed of the ball, relative to the wagon at the instant it hits a wall ?Q. 96. Five particles are projected simultaneously from the top of a tower that is
h
 = 32
m
 high. The initial velocities of projection are as shown in figure. Velocity of 2 and 5 are horizontal.
12234553°37°37°10 m/s10 m/s10 m/s m/s15 m/s
h 
= 32m
 (a) Which particle will hit the ground first? (b) Separation between which two particles is maximum at the instant the first particle hits the ground? (c) Which two particles are last and last but one to hit the ground? Calculate the distance between these two particles (still in air), at a time 0.3
s
 after the third particle lands on ground. [
g
 = 10
m/s
2
,
tan37 34
° =
]
 
Q. 97. From the top of a long smooth incline a small body
 A
 is projected along the surface with speed
u
. Simultaneously, another small object
 B
 is thrown horizontally with velocity
v
 = 10
m
 / 
s
, from the same point. The two bodies travel in the same vertical plane and body
 B
 hits body
 A
 on the incline. If the inclination angle of the incline is
θ 
 =     
−
cos
 1
 45
 find (a) the speed
u
 with which
 A
 was projected. (b) the distance from the point of projection, where the two bodies collide.
4
 
 2.16
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
 
B VA
u 
q 
 
Q. 98. A man is on straight road
 AC 
, standing at
 A
. He wants to get to a point
P
 which is in field at a distance ‘
d 
’ off the road (see figure). Distance
 AB
 is
l
 = 50. The man can run on the road at a speed
v
1
 = 5
m
 / 
s
 and his speed in the field is
v
2
 = 3
m
 / 
s
.
A B CP
d l 
 (a) Find the minimum value of ‘
d 
’ for which man can reach point
P
 in least possible time by travelling only in the field along the straight line
 AP
. (b) If value of ‘
d 
’ is half the value found in (a), what length the man must run on the road before entering the field, in order to reach ‘
P
’ in least possible time.
 
Q. 99. Two particles,
 A
 and
 B
 are moving in concentric circles in anticlockwise sense in the same plane with radii of the circles being
g 
 A
 = 1.0
m
 and
g 
 B
 = 2.0
m
respectively. The particles move with same angular speed of
w 
 = 4
rad 
 / 
s
. Find the angular velocity of
 B
 as observed by
 A
 if  (a) Particles lie on a line passing through the centre of the circle. (b) Particles lie on two perpendicular lines passing through the centre.
 
Q. 100. (a) An unpowered rocket is in flight in air. At a moment the tracking radar gives following data regarding the rocket. 
r 
 = distance of the rocket from the radar = 4000
m
,
dr dt d dt 
=
 0,
 θ 
 = 1.8 deg/sec; where
q 
 is the angle made by position vector of the rocket with respect to the vertical.
q 
Rocket
r 
y x 
 (a) Neglect atmospheric resistance and take g = 9.8
m
 / 
s
2
 at the concerned height. Neglect height of radar. Calculate the height of the rocket above the ground. (b) Two points A and B are moving in
 X 
 -
Y 
 plane with constant velocity of
V i j
 A
 = -
( )
69ˆˆ
 
m/s
 and
V i j
 B
 = +
( )
ˆˆ 
m/s
respectively. At time
t 
 = 0 they are 15
m
 apart and both of them lie on
 y
 axis with
 A
 lying away on positive
Y 
 axis with respect to
 B
. What is the angular velocity of
 A
 with respect to
 B
 at
t 
 = 1
s
? Q. 101. A stone is projected horizontally with speed
u
 from the top of a tower of height
h.
 (a) Calculate the radius of curvature of the path of the stone at the point where its tangential and radial accelerations are equal. (b) What shall be the height (
h
) of the tower so that radius of curvature of the path is always less than the value obtained in (
a
) above.Q. 102. A stick of length
 L
 = 2.0
m
 is leaned against a wall as shown. It is released from a position when
q 
 = 60°. The end
 A
 of the stick remains in contact with the wall and its other end
 B
 remains in contact with the floor as the stick slides down. Find the distance travelled by the centre of the stick by the time it hits the floor.
AB
q 
2
 
 K
INEMATICS
 
2.17
Q. 103. (a) A line
PQ
 is moving on a fixed circle of radius
 R
. The line has a constant velocity
v
perpendicular to itself. Find the speed of point of intersection (
 A
) of the line with the circle at the moment the line is at a distance
d 
 =
 R
 /2 from the centre of the circle.
AP d Q 
 (b) In the figure shown a pin
P
 is confined to move in a fixed circular slot of radius
 R
. The pin is also constrained to remains inside the slot in a straight arm
O
'
 A
. The arm moves with a constant angular speed
w 
 about the hinge
O
'. What is the acceleration of point
P
?
BAPOO'CR R
 
Q. 104. A flexible inextensible cord supports a mass
 M
as shown in figure.
 A
1
,
 A
2
 and
 B
 are small pulleys in contact with the cord. At time
t 
 = 0 cord
PQ
 is horizontal and
 A
1
,
 A
2
 start moving vertically down at a constant speed of
v
1
, whereas
 B
 moves up at a constant speed of
v
2
. Find the velocity of mass
 M 
 as a function of time.
LPA
1
 A
2
QMBL L L
 Q. 105. In the arrangement shown in the figure
 A
 is an equilateral wedge and the ball
 B
 is rolling down the incline
 XO
. Find the velocity of the wedge (of course, along
OY 
) at the moment velocity of the ball is 10
m
 / 
s
 parallel to the incline
 XO
.
y 
60°30°OB
1  0  m   /   s  
 Q. 106. A meter stick
 AB
 is lying on a horizontal table. Its end
 A
 is pulled up so as to move it with a constant velocity
V 
 A
 = 4
ms
–1
 along a vertical line. End
 B
 slides along the floor.
V 
A
AB 
 (a) After how much time (
t 
0
) speed (
V 
 B
) of end B becomes equal to the speed (
V 
 A
) of end A ? (b) Find distance travelled by the end
 B
 in time
t 
0
. Q. 107.
 
One end of a rope is fixed at a point on the ceiling the other end is held close to the first end so that the rope is folded. The second end is released from this position. Find the speed at which the fold at
F 
 is descending at the instant the free end of the rope is going down at speed
V 
.
V F 
 Q. 108. Block
 A
 rests on inclined surface of wedge
 B
 which rests on a horizontal surface. The block
 A
 is connected to a string, which passes over a pulley
P
 (fixed rigidly to the wedge
 B
) and its other end is securely fixed to a wall at
Q
. Segment
PQ
 of the string is horizontal and
Q
 is at a large distance
5
 
 2.18
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
from
P
. The system is let go from rest and the wedge slides to right as
 A
 moves on its inclined face. Find the distance travelled by
 A
 by the time it reaches the bottom of the inclined surface.
PQBCA
q 
= 30°5m
 
Q. 109. Two frictionless ropes connect points
 A
 &
 B
 in vertical plane. Bead 1 is allowed to slide along the straight rope
 AB
 and bead 2 slides along the curved rope
 ACB
. Which bead will reach
 B
 in less time?
A1B2C
L 
EVEL 
 3
Q. 110. A car manufacturer usually tells a optimum speed (
V 
0
) at which the car should be driven to get maximum mileage. In order to find the optimum speed for a new model, an engineer of the car company experimented a lot and finally plotted a graph between the extreme time
t 
 (defined as number of hours a tank full of petrol lasts) vs the constant speed
V 
 at which car was run.
t 
 (hour)2043O10 150V(km hr )
-1
 (a) Calculate the optimum speed
V 
0
 for this new model. (b) If the fuel tank capacity of this car is 50 litre, what maximum mileage can be obtained from this car? Q. 111. While starting from a station, a train driver was instructed to stop his train after time
T 
 and to cover maximum possible distance in that time. (a) If the maximum acceleration and retardation for the train are both equal to ‘
a
’, find the maximum distance it can cover. (b) Will the train travel more distance if maximum acceleration is ‘
a
’ but the maximum retardation caused by the brakes is ‘2
a
’? Find this distance. Q. 112. Two particles 1 and 2 start simultaneously from origin and move along the positive
 X 
 direction. Initial velocity of both particles is zero. The acceleration of the two particles depends on their displacement (
 x 
) as shown in fig.
O
a
1
2
 0
aa
0
X
0
X O
a
2
2
 0
aa
0
X
0
X
 (a) Particles 1 and 2 take
t 
1
 and
t 
2
 time respectively for their displacement to become
 x 
0
. Find
21
. (b) Which particle will cover 2
 x 
0
 distance in least time? Which particle will cross the point
 x 
 = 2
 x 
0
 with greater speed? (c) The two particles have same speed at a certain time after the start. Calculate this common speed in terms of
a
0
 and
 x 
0
.Q. 113. A cat is following a rat. The rat is running with a constant velocity
u
. The cat moves with constant speed
v
 with her velocity always directed towards the rat. Consider time to be
t 
 = 0 at an instant when both are moving perpendicular to each other and separation between them is
 L
. (a) Find acceleration of the cat at
t 
 = 0. (b) Find the time
t 
0
 when the rat is caught. (c) Find the acceleration of the cat immediately before it catches the rat. (d) Draw the path of the rat as seen by the cat. Q. 114.(a) Prove that bodies starting at the same time
t 
 = 0 from the same point, and following frictionless slopes in different directions in the same vertical plane, all lie in a circle at any subsequent time.
5
 
 K
INEMATICS
 
2.19
 (b) Using the above result do the following problem.
 A
 point
P
 lies above an inclined plane of inclination angle
a 
.
P
 is joined to the plane at number of points by smooth wires, running in all possible directions. Small bodies (in shape of beads) are released from
P
 along all the wires simultaneously. Which body will take least time to reach the plane. 
P
a 
Q. 115. The acceleration due to gravity near the surface of the earth is
g
. A ball is projected with velocity
u
 from the ground. (a) Express the time of flight of the ball. (b) Write the expression of average velocity of the ball for its entire duration of flight. Express both answers in terms of
u
 and
g
.Q. 116. A ball is projected from point
O
 on the ground. It hits a smooth vertical wall
 AB
at a height
h
 and rebounds elastically. The ball finally lands at a point
C 
 on the ground. During the course of motion, the maximum height attained by the ball is
 H 
.
BAOC
 (a) Find the ratio
h H 
 if
OAOC 
13
 (b) Find the magnitude of average acceleration of the projectile for its entire course of flight if it was projected at an angle of 45° to the horizontal. Q. 117. A boy can throw a ball up to a speed of
u
 = 30
m
 / 
s
 . He throws the ball many a times, ensuring that maximum height attained by the ball in each throw is
h
 = 20
m
. Calculate the maximum horizontal distance at which a ball might have landed from the point of projection. Neglect the height of the boy. [
g
 = 10
m
 / 
s
2
] Q. 118. A valley has two walls inclined at 37° and 53° to the horizontal. A particle is projected from point
P
 with a velocity of
u
 = 20
m
 / 
s
 along a direction perpendicular to the incline wall
OA
. The Particle hits the incline surface RB perpendicularly at Q. Take
g
 = 10
m
 / 
s
2
 and find: (a) The time of flight of the particle. (b) Vertical height h of the point
P
 from horizontal surface
OR
. 
tan37 34
° =
u h 
37°R53°QBAO
 Q. 119.
h 
 
A ball is released in air above an incline plane inclined at an angle
a 
 to the horizontal. After falling vertically through a distance
h
 it hits the incline and rebounds. The ball flies in air and then again makes an impact with the incline. This way the ball rebounds multiple times. Assume that collisions are elastic, i.e., the ball rebound without any loss in speed and in accordance to the law of reflection. (a) Distance between the points on the incline where the ball makes first and second impact is
l
1
 and distance between points where the ball makes second and third impact is
l
2
. Which is large
l
1
 or
l
2
? (b) Calculate the distance between the points on the incline where the ball makes second and fifth impact.
 
Q. 120. A terrorist ‘
 A
’ is walking at a constant speed of 7.5
km
 / 
hr 
 due West. At time
t 
 = 0, he was exactly
3
 
 2.20
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
South of an army camp at a distance of 1 km. At this instant a large number of army men scattered in every possible direction from their camp in search of the terrorist. Each army person walked in a straight line at a constant speed of 6
km
 / 
hr 
. (a) What will be the closest distance of an army person from the terrorist in this search operation? (b) At what time will the terrorist get nearest to an army person?Q. 121. A large wedge
 BCD
, having its inclined surface at an angle
q 
 = 45° to the horizontal, is travelling horizontally leftwards with uniform velocity
u
 = 10
m
 / 
s
V C D B 
 45°
A
 At some instant a particle is projected vertically up with speed
V 
 = 20
m
 / 
s
 from point A on ground lying at some distance right to the lower edge B of the wedge. The particle strikes the incline BC normally, while it was falling. [
g
 = 10
m
 / 
s
2
] (a) Find the distance
 AB
 at the instant the particle was projected from
 A
. (b) Find the distance of lower edge
 B
 of the wedge from point
 A
 at the instant the particle strikes the incline. (c) Trace the path of the particle in the reference frame attached to the wedge. Q. 122. The speed of river current close to banks is nearly zero. The current speed increases linearly from the banks to become maximum (=
V 
0
) in the middle of the river. A boat has speed ‘
u
’ in still water. It starts from one bank and crosses the river. Its velocity relative to water is always kept perpendicular to the current. Find the distance through which the boat will get carried away by the current (along the direction of flow) while it crosses the river. Width of the river is
l
.Q. 123. A water sprinkler is positioned at
O
 on horizontal ground. It issues water drops in every possible direction with fixed speed
u
. This way the sprinkler is able to completely wet a circular area of the ground (see fig). A horizontal wind starts blowing at a speed of . Mark the area on the ground that the sprinkler will now be able to wet.
O
Q. 124. A cylinder of radius R has been placed in a corner as shown in the fig. A wedge is pressed against the cylinder such that its inclined surfaces touches the cylinder at a height of
25
 from the ground. Now the wedge is pushed to the left at a constant speed
V 
 = 15
m
 / 
s
. With what speed will the cylinder move?
R V 2R/5 
 Q. 125. The entrance to a harbour consists of 50
m
 gap between two points
 A
 and
 B
 such that
 B
 is due east of
 A
. Outside the harbour there is a 8 km/hr current flowing due east. A motor boat is located 300 m due south of A. Neglect size of the boat for answering following questions- (a) Calculate the least speed (
V 
min
) that the motor boat must maintain to enter the harbour. (b) Show that the course it must steer when moving at
V 
min
 does not depend on the speed of the current. Q. 126. Two small pegs (
 A
 and
 B
) are at horizontal and vertical separation
b
 and
h
 respectively. A small block of mass
 M 
 is suspended with the help of two light strings passing over
 A
 and
 B
 as shown in fig. The two string are always kept at right angles (i.e., <
 APB
 = 90°). Find the minimum possible gravitation potential energy of the mass assuming the reference level at location of peg
 A
. [Hint: the potential energy is minimum when the block is at
3
 
 K
INEMATICS
 
2.21
its lowest position]
hAPMB
b 
 Q. 127. (a) A canon fires a shell up on an inclined plane. Prove that in order to maximize the range along the incline the shell should be fired in a direction bisecting the angle between the incline and the vertical. Assume that the shell fires at same speed all the time. (b) A canon is used to hit a target a distance
 R
 up an inclined plane. Assume that the energy used to fire the projectile is proportional to square of its projection speed. Prove that the angle at which the shell shall be fired to hit the target but use the least amount of energy is same as the angle found in part (a)
u
Q. 128. A ball of mass
m
 is projected from ground making an angle
q 
 to the horizontal. There is a horizontal wind blowing in the direction of motion of the ball. Due to wind the ball experiences a constant horizontal force of
mg 
 in direction of its motion. Find
q
for which the horizontal range of the ball will be maximum.Q. 129. A projectile is projected from a level ground making an angle
q 
 with the horizontal (
 x 
 direction). The vertical (
 y
) component of its velocity changes with its
 x 
 co-ordinate according to the graph shown in figure. Calculate
q 
. Take
g
 = 10
ms
–2
.
(In )
m 
O45° 10
x v m s 
y 
 Q. 130. In the arrangement shown in the figure, the block
C 
 begins to move down at a constant speed of 7.5
cm
 / 
s
 at time
t 
 = 0. At the same instant block
 A
 is made to start moving down at constant acceleration. It starts at
 M 
 and its speed is 30
cm
 / 
s
 when it reaches
 N 
 (
 MN 
 = 20
cm
)
.
Assuming that
 B
 started from rest, find its position, velocity and acceleration when block
 A
 reaches
 N 
.
ACBBMN20 cm
 
Q. 131. A rocket prototype is fired from ground at time
t 
 = 0 and it goes straight up. Take the launch point as origin and vertically upward direction as positive
 x 
 direction. The acceleration of the rocket is given by
a gkt t t g t  t 
= - < £= - >
20
200
;;
 Where
t  g k 
0
32
 (a) Find maximum velocity of the rocket during the up journey. (b) Find maximum height attained by the rocket. (c) Find total time of flight.
 
Q. 132. A man standing inside a room of length
 L
 rolls a ball along the floor at time
t
= 0. The ball travels at constant speed
v
 relative to the floor, hits the front wall (
 B
) and rebounds back with same speed
v
. The man catches the ball back at the wall
 A
 at time
t 
0
. The ball travelled along a straight line relative to the man inside the room. Another observer standing outside the room found that the entire room was travelling horizontally at constant velocity
v
 in a direction parallel to the
3
 
 2.22
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
two walls
 A
 and
 B
.
Lv v A
 (a) Find the average speed of the ball in the time interval
t 
 = 0 to
t 
 =
t 
0
 as observed by the observer outside the room. (b) If the room has acceleration in the direction of its velocity draw a sketch of the path of the ball as observed by the observer standing outside. Assume that velocity of room was
v
 at the instant the ball was released.
 
Q. 133. There is a tall cylindrical building standing in a field. Radius of the cylinder is
 R
 = 8
m
. A boy standing at
 A
 (at a distance of 10
m
 from the centre of the cylindrical base of the building) knows that his friend is standing at
 B
 behind the building. The line joining
 A
 and
 B
 passes through the centre of the base of the building. Distance between
 A
 and
 B
 is 50
m
. A wants to throw a ball to
 B
 but he realizes that the building is too tall and he cannot throw the ball over it. He throws the ball at a speed of 20
m/s
 such that his friend at
 B
 has to move minimum distance to catch it.
10 mAB40 m
R 
= 8mO
 (a) What is the minimum distance that boy at
 B
 will have to move to catch the ball? (b) At what angle to the horizontal does the boy at
 A
 throws the ball? Assume that the ball is released and caught at same height above the ground. [Take
g
 = 10
m
 / 
s
2
 and sin
–1
 (0.75) ~ 48.6° Q. 134. A wet umbrella is held upright (see figure). The man holding it is rotating it about its vertical shaft at an angular speed of
w 
 = 5
rad 
 
s
–1
. The rim of the umbrella has a radius of
r 
 = 0.5
m
 and it is at a height of
 H 
 = 1.8
m
 from the floor. The man holding the umbrella gradually increases the angular speed to make it 2
w 
. Calculate the area of the floor that will get wet due to water drops spun off the rim and hitting the floor. [
g
 = 10
m
 / 
s
2
]
r 
= 0.5 m
H m 
= 1.8
 
Q. 135.
 
A ball is projected vertically up from ground. Boy
 A
 standing at the window of first floor of a nearby building observes that the time interval between the ball crossing him while going up and the ball crossing him while going down is
t 
1
. Another boy
 B
 standing on the second floor notices that time interval between the ball passing him twice (during up motion and down motion) is
t 
2
. (a) Calculate the height difference (
h
) between the boy
 B
 and
 A.
 (b) Assume that the height of boy
 A
 from the point of projection of the ball is also equal to
h
 and calculate the speed with which the ball was projected. Q. 136. A stick of length
 L
 is dropped from a high tower. An ant sitting at the lower end of the stick begins to crawl up at the instant the stick is released. Velocity of the ant relative to the stick remains constant and is equal to
u
. Assume that the stick remains vertical during its fall, and length of the stick is sufficiently long.
L 
4
 
 K
INEMATICS
 
2.23
 (a) Calculate the maximum height attained by the ant measured from its initial position. (b) What time after the start the ant will be at the same height from where it started?
 
Q. 137. Two balls are projected simultaneously from the top of a tall building. The first ball is projected horizontally at speed
u
1
 = 10
m
 / 
s
 and the other one is projected at an angle
θ 
 =     
−
tan
 1
 43
 to the horizontal with a velocity
u
2
. [g = 10
m
 / 
s
2
]
2
u 
1
q 
 (a) Find minimum value of
u
2
 (=
u
0
) so that the velocity vector of the two balls can get perpendicular to each other at some point of time during their course of flight. (b) Find the time after which velocities of the two balls become perpendicular if the second one was projected with speed
u
0
.
 
Q. 138. There is a large wedge placed on a horizontal surface with its incline face making an angle of 37° to the horizontal. A particle is projected in vertically upward direction with a velocity of
u
 = 6.5
m
 / 
s
 from a point
O
 on the inclined surface. At the instant the particle is projected, the wedge begins to move horizontally with a constant acceleration of
a
 = 4
m
 / 
s
2
. At what distance from point
O
will the particle hit the incline surface if (i) direction of
a
 is along
 BC 
? (ii) direction of
a
 is along
 AB
?
OC BA37°
 
Q. 139. The windshield of a truck is inclined at 37° to the horizontal. The truck is moving horizontally with a constant acceleration of
a
 = 5
m
 / 
s
2
. At the instant the velocity of the truck is
v
0
 = 0.77
m
 / 
s
, an insect jumps from point
 A
 on the windshield, with a velocity
u
 = 2.64
m
 / 
s
 (relative to ground) in vertically upward direction. It falls back at point
 B
 on the windshield. Calculate distance
 AB
. Assume that the insect moves freely under gravity and
g
 = 10
m
 / 
s
2.
AB
a m/s 
= 5
2
37°
 
Q. 140. Two persons are pulling a heavy block with the help of horizontal inextensible strings. At the instant shown, the velocities of the two persons are
v
1
 and
v
2
directed along the respective strings with the strings making an angle of 60° between them. (a) Find the speed of the block at the instant shown. (b) For what ratio of
v
1
 and
v
2
 the instantaneous velocity of the block will be along the direction of
v
1
.
1
v 
2
60°
Q. 141. A heavy block '
 B
' is sliding with constant velocity
u
 on a horizontal table. The width of the block is
 L
. There is an insect
 A
 at a distance
d 
 from the block as shown in the figure. The insect wants to cross to the opposite side of the table. It begins to crawl at a constant velocity
v
 at the instant shown in the figure. Find the least value of
v
 for which the insect can cross to the other side without getting hit by the block. 
u B Ld A
4
 
 2.24
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
Q. 142. A projectile is thrown from ground at a speed
v
0
at an angle
a 
 to the horizontal. Consider point of projection as origin, horizontal direction as
 X 
 axis and vertically upward as
Y 
 axis. Let
t 
 be the time when the velocity vector of the projectile becomes perpendicular to its position vector. (a) Write a quadratic equation in
t 
. (b) What is the maximum angle
a 
 for which the distance of projectile from the point of projection always keeps on increasing? [Hint: Start from the equation you obtained in part (a)]A projectile is thrown from a point on ground, with initial velocity
u
 at some angle to the horizontal. Show that it can clear a pole of height
h
 at a distance
d 
 from the point of projection if 
u
2
 >
g h h d 
[]
+ +
22
 A particle rotates in a circle with angular speed
w 
0
. A retarding force decelerates it such that angular deceleration is always proportional to square root of angular velocity. Find the mean angular velocity of the particle averaged over the whole time of rotation.
ANSWERS
 The two velocities are perpendicular. (a) 7.5
km
 / 
hr 
–1
 (b) 2
hr 
 40 min
3.
 (a)
F 
 (b)
T 
 (c)
T 
4.
 (a)
 E 
, (b)
 D
,
G
 (c)
 B
,
C 
5.
 (a) 4
m
 / 
s
 (b) 2
m
 / 
s
2
 (a)
 X 
max
 = 4
m
 ;
t 
 = 2
s
 (b)
84
 t s 
( )
x m 
( )
 
t t v v 
0
–
v 
0
 (i)
 B
 and
C 
 (ii)
 D
 (iii)
 A
,
 B
,
C 
,
 D
 10
mv
 = 12
m
 / 
s
 
v 
 (m/s)30
t 
O104 8
12.
 (a) particle
 A
 (b) see solution for graph (a) l <
t 
 < 2
s
 and 3 <
t 
 < 4
s
 (b) 1
m
 / 
s
14.
 (a) 22 (
Km
) (
s
–1
) (
 MLy
–1
) (b)
1 2n( )
 (a)
20
V (m / s) 
O2 4
t (s) 
–20
5
 
 K
INEMATICS
 
2.25
 (b)
20
V (m / s) 
O 2 4
t (s) 
 (c)
h (m) 
20 mO24
t (s) 
 (d)
Distance (m)40O4
t (s) 
 
y H O V 
0 
V H / 2 
  D  o  w  n   m  o  t  i  o  n
U    p  m  o  t  i   o  n  
V 
0
2
 
V 
 = 20
ms
–1
∆
t  L g 
= − 
22 1
 1
m
20
. All statements are true 
y x 
40
m
 The one that is projected at
q 
2
 
1221
=
tantan
θ θ 
 20 (1 + 2)
m
24.
 (a) 12.13
m
 (b) 16
m
 / 
s
 
a
 = 5.19
m
 / 
s
2
 
u
= 16
m/s
;
θ 
 =     
−
tan
 1
 4 215
 
45
 43
p 
 R
 Displacement = 40
cm
 Distance = (30 5 + 10 13)
cm
 4041
s
31.
 (a)
t 
0
 = 5
s
 (b) car 4 1
m
 / 
s
2
 to 7
m
 / 
s
2
 (a)
V 
max
= 12
km
 / 
hr 
 (b) 
cmin 
0°90°
 
 5
m
 / 
s
, 12
m
 / 
s
 
D
t 
 = 23.33
s
 
 Lu
 807
kph
38.
 (a) Parabolic path (b) 6
m
 / 
s
 (a) 12
Tg
cos
q 
 Perpendicular to the incline (b) 4 2
ms
–1
 2 2
cms
–1
 883min 
a
t 
 = 0; path is circular 
n  w q 
=
 H 
1
 
 2.26
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
(a)
a a
=
3
0
p 
 (b) 8.37 m/s
2
45.
(a)
x 
t –v v 
 (b)
a
y 
t 
(a)
r 
 =
vt 
 [cos (
w 
t 
)ˆ
i
 + sin (
w 
t 
)]ˆ
 j
 (b)
V V t t t 
 p
    
= -
[cos()sin()]
w w w 
 ˆ
i
+
V 
 [sin (
w 
t 
) +
w 
t 
 cos (
w 
t 
)]ˆ
 j
 (a) v
0
e
2
p 
n
 (b)
V 
 = 2
a
0
 x 
 3
u
, zero 38 
rad 
 2
m
 / 
s
 1.59
s
52.
 (a)
t s
0
323
=
; (b)
< > = < > =
V V m s
 A B
1538
π 
 / 
53.
 Both are true60
s
55.
(a) Harshit (b)
t 
    H   A    R   S    H    I    T   A    K   A    N    K   S    H   A
O AC D B x t 
H 
t 
H 
2
t 
A
 (c) 2
uvu v
 (d)
2
 Lu v
56.
 (b) 0.5
m
57.
 (a) 10 5
s
 (b)
x AB t 
1
10O
 t 
2 
t t(s) 
58.
(a) Acceleration is increasing (b) 1
m
 / 
s
2
59.
∆
r yi xj d 
= − +
( )
ˆˆ
θ 
60.
(a) 8.5
s 
(b) 2.41
m
61.
t 
 = (2 – 2)
t 
0
 ;
 x 
 = (2 – 1)
 x 
0
62.
2.2
hr
; 90.9
km
 / 
hr 
63.
66
m
64.
θ 
 =     
−
tan
1
32
65.
 (a) 48
m
, 68.5
m
 (b)
x(m) t(m) 
60.552.55.5 7.5O
66.
34
02
v L
5
 
 K
INEMATICS
 
2.27
67.
 
s (m) 
5O1 2 3 4
t(s) 
68.
 (i)
v
10
 =
v
20
 = 4
m
 / 
s
 (ii) Both will take same time
69.
 (a) 90
m
 (b)
V 
–304
t (s) 
 (c)
O V (m/s) 
– 45
t 
4
70.
(a) 2.5
s
 (b)
6.255.51.5O
y 
 (In meter)2.0 2.5
 t 
 
71.
 (a)
t u
0
2
=
α 
 (b)
23
3 21 2
( )
 //
α 
 (c)
72.
 (a) 2 (b) zero
73.
 
v 
10
 m/s t s 
( )
s m 
( )5O
 t s 
( )
 
 Ak 
0
v
0
2
75.
 (a)
α 
 =     
−
tan
 1
 34
 (b)
θ 
 =
−
tan
 1
 12
 (c) The ball will hit at a point lower than the earlier spot.
76.
 (i)
n
 = 84° (ii)
u g 
77.
 (a) 80
m
 (b) 1.6 × 10
–3
78.
 (a) 
q 
 = 60° (b) 3 
u
79.
 202
m
80.
 (a) 11.25
m
 (b)
tan
−
     
1
 85
81.
 (a)
 L
 = 14.58
m
 (b)
OB
= 41.66
m
82.
(a) 5
m
 (b) 480
m
 
u
 = 7.29
m
 / 
s
,
t 
 = 1
s
.
84.
 (a)
h g 
 (b)
u
 = 21
gR
3
 
 2.28
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
85.
 (a) No (b)
Q O  V 
86.
40158989ˆ.ˆ.ˆ
i j k 
+ -
( )
87.
 (a)
q 
 = tan
–1
 (2) (b) 60 5
kmhr
–1
88.
 (a) 34
hr 
 (b) 4.5
km
89.
 (a) 45° (b) 2
m
 / 
s
90.
 (a)
ug
2
2sincos
a q 
 (b)
u
cos()cos
a q q 
+
91.
 (b)
92.
 (a) yes, 2
s
 (b) zero (c) straight line
93.
 733
0
V 
 
94.
 
 x  x v uv u
12
=+−
95.
 (a) Corner C (b)
u
96.
 (a) particle 1 (b) Particle 2 and 5 (c) particle 3 and 4 ; 50.94
m
97.
 (a)
u
 = 8
m
 / 
s
, (b) 18.75
m
98.
 (a)
d 
min
2003 (b) 25
m
99.
 (a)
w 
 = 4
rad 
 / 
s 
(b)
w 
 = 4
rad 
 / 
s
100.
 (a) 1600
m
 (b) 32
rad 
 /sec 
101.
 (a)
 R ug
=
22
2
 (b)
h ug
2
2
102.
 
p 
3
m
103.
 (a) 23
v
 (b) 4
w 
2
 R
104.
 
v dydt v t  L v t v v t  L v v t 
= =+++
( )
+ +
( )
22
12212212221222
105.
 103 
m/s
106.
 (a)
t s
0
142
=
 (b)
112
-Ê Ë Áˆ ¯ ˜ 
 m
107.
 
V 
 /2
108.
 10 sin 15°
109.
 Bead 2
110.
 (a) 80
kmhr
–1
 (b) 17
kml
–1
111.
 (a) 14
2
aT 
 (b) yes,
13
2
aT 
112.
 (a) 2 (b) particle 1 will cover 2
 x 
0
 in lesser time. Both will cross 2
 x 
0
 with same speed. (c)
v
 = (2 + 2)
a
0
 x 
0
113.
 (a)
uv L
 (b)
t  vLv u
0 2 2
=−
 (c) Zero (d) The path will be like a spiral
114.
 (b) Body travelling along a line making an angle
4
 
 K
INEMATICS
 
2.29
with vertical
115.
 (a)
t u g  g 
= −
2
2
       
.
 (b)
V u g u g  g 
av
          
= −
( )
.
2
116.
 (a)
1625
 (b) 2
g
117.
 40 5
m
118.
 (a) 2.5
s
 (b) 4.05
m
119.
 (a)
l
2
 >
l
1
 (b) 72
h
 sin
a 
120.
 (a)
35
km
 (b) 8 min
121.
 (a) 15
m
 (b) 15
m 
(c) parabolic
122.
 
V l u
0
2
123.
 A circle of same size shifted from the original circle by
∆
 X  ug
=
2
2 in the direction of wind.
124.
 20
m
 / 
s
125.
 (a)
4837
km hr 
/
126.
 
U Mg h b h
min
 = − + − 
12
2 2
128.
 
q 
 = 60°
129.
 
q 
 = 45°
130.
 Position: 40
cm
 up from starting position 
V 
 B
 = 45
cm
 / 
s
 
≠
( )
 
a
 B
 = 22.5
cm
 / 
s
2
 
≠
( )
131.
 (a)
V  g k 
max3
18
 (b)
 X  g k 
02
316
 (c)
T  g k 
3232
132.
 (a) 2
v
 (b) path is as shown 
133.
 (a) 40
m
 (b) 24.3° or 65.7°
134.
 21.2
m
2
135.
 (a)
h g t t 
=−
( )
1222
8
 (b)
u g t t 
= −
22
1222
136.
 (a)
 H  u g 
max2
2
 (b)
u g 
137.
 (a)
u
0
 = 37.5
m
 / 
s
 (b)
t 
 = 1.5
m
 / 
s
138.
 (i) 3.38
m
 (ii) 2.5
m
139.
 
 AB
 = 0.57
m
140.
 (a)
23
 12221 2
+ −
 (b)
12
2
141.
 
v uLd L
min
 =+
2 2
142.
 (a)
t  vgt  vg
20022
320
− + =
sin
α 
 (b)
sin
 1
 89
144.
 
w 
0
3
4
 
NEWTON’S LAWS
L 
EVEL 
 1
 
Q. 1. Let
u
 be the initial velocity of a particle and
F 
 be the resultant force acting on it. Describe the path that the particle can take if (a)
u F 
¥ =
0 and
F 
 = constant (b)
u F 
.
 =
0 and
F 
=
 constant In which case can the particle retrace its path.
 
Q. 2. A ball is projected vertically up from the floor of a room. The ball experiences air resistance that is proportional to speed of the ball. Just before hitting the ceiling the speed of the ball is 10
m
 / 
s
 and its retardation is 2
g
. The ball rebounds from the ceiling without any loss of speed and falls on the floor 2
s
 after making impact with the ceiling. How high is the ceiling? Take
g
= 10
m/s
2
.
 
Q. 3. A small body of super dense material, whose mass is half the mass of the earth (but whose size is very small compared to the size of the earth), starts from rest at a height
 H 
 above the earth’s surface, and reaches the earth’s surface in time
t 
. Calculate time
 t 
 assuming that
 H 
 is very small compared to the radius of the earth. Acceleration due to gravity near the surface of the earth is
g
.
 
Q. 4.
 N 
 identical carts are connected to each other using strings of negligible mass. A pulling force
F 
 is applied on the first cart and the system moves without friction along the horizontal ground. The tension in the string connecting 4
th
 and 5
th
 cart is twice the tension in the string connecting 8
th
 and 9
th
 cart. Find the total number of carts (
 N 
) and tension in the last string.
F N N– 
1 12
 
Q. 5. A toy cart has mass of 4
kg
 and is kept on a smooth horizontal surface. Four blocks
 A, B, C 
 and
 D
 of masses 2
kg
, 2
kg
, 1
kg
 and 1
kg
 respectively have been placed on the cart.
 A
 horizontal force of
F
= 40
 N 
 is applied to the block
 A
 (see figure). Find the contact force between block
 D
 and the front vertical wall of the cart.
F AB C D 
 Q. 6. (i) Three blocks
 A, B
 and
C 
 are placed in an ideal Atwood machine as shown in the figure. When the system is allowed to move freely it was found that tension in the string connecting
 A
 to
C 
 was more than thrice the tension in the string connecting
 A
 and
 B
. The masses of the three blocks
 A, B
 and
C 
 are
m
1
,
m
2
 and
m
3
, respectively. State whether the following statements are true or false [All masses have finite non zero values and the system has a non zero acceleration].
AB C m 
1
m 
2
m 
3
 (a)
m
3
 can have any finite value (b)
m
1
 > 2
m
2
 (ii) In an Atwood machine the sum of two masses is a constant. If the string can sustain a tension equal to
2430
Ê Ë Á ˆ ¯ ˜ 
 of the weight of the sum of two masses, find the least acceleration of the masses. The string and pulley are light.
03
5
 
 3.2
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
 (iii) A load of
w
 newton is to be raised vertically through a height
h
 using a light rope. The greatest tension that the rope can bear is
h
w
(
h
 
> 1). Calculate the least time of ascent if it is required that the load starts from rest and must come to rest when it reaches a height
h
.Q. 7. In the arrangement shown in the figure the system is in equilibrium. Mass of the block
 A
 is
 M 
 and that of the insect clinging to block
 B
 is
m
. Pulley and string are light. The insect loses contact with the block
 B
 and begins to fall. After how much time the insect and the block B will have a separation
 L
 between them.
A B 
 
Q. 8. Two blocks of equal mass,
 M 
 each, are connected to two ends of a massless string passing over a massless pulley. On one side of the string there is a bead of mass
 M 
2.
M 
2
M M 
 (a) When the system is released from rest the bead continues to remain at rest while the two blocks accelerate. Find the acceleration of the blocks. (b) Find the acceleration of the two blocks if it was observed that the bead was sliding down with a constant velocity relative to the string. Q. 9. A pulley is mounted on a stand which is placed over a weighing scale. The combined mass of the stand and the pulley is
 M 
0
. A light string passes over the smooth pulley and two masses
m
 and
 M 
 (>
m
) are connected to its ends (see figure). Find the reading of the scale when the two masses are left free to move.
Stand
M 
0
m  M 
 Q. 10. In the given arrangement, all strings and pulleys are light. When the system was released it was observed that
 M 
 and
m
0
 do not move. Find the masses
 M 
 and
m
0
 in terms of
m
1
 and
m
2
. Find the acceleration of all the masses if string is cut just above
m
2
.
m 
0
m 
1
m 
2
M 
 Q.11 The system shown in the fig. is in equilibrium. Pulleys
 A
 and
 B
 have mass
 M 
 each and the block C has mass 2
 M 
. The strings are light. There is an insect (D) of mass
 M 
 /2 sitting at the middle or the right string. Insect does not move.
CDS1AES2B
5
 
 N
EWTON
’
S
 L
 AWS
 
3.3
 (a) Just by inspection, say if the tension in the string S1 is equal to, more than or less than 9/2
 Mg
. (b) Find tension in the string
S 
2, and
S 
1. (c) Find tension in
S 
2 if the insect flies and sits at point E on the string. Q. 12. A block slides down a frictionless plane inclined at an angle
q 
. For what value of angle
q 
 the horizontal component of acceleration of the block is maximum? Find this maximum horizontal acceleration. Q. 13. A tall elevator is going up with an acceleration of
a
 = 4
m/s
2
. A 4
kg
 snake is climbing up the vertical wall of the elevator with an acceleration of
a
. A 50 g insect is riding on the back of the snake and it is moving up relative to the snake at an acceleration of
a
. Find the friction force between the elevator wall and the snake. Assume that the snake remains straight. Q. 14. Due to air drag the falling bodies usually acquire a constant speed when the drag force becomes equal to weight. Two bodies, of identical shape, experience air drag force proportional to square of their speed (
F 
drag
 =
kv
2
,
k 
 is a constant). The mass ratio of two bodies is 1 : 4. Both are simultaneously released from a large height and very quickly acquire their terminal speeds. If the lighter body reaches the ground in 25
s
, find the approximate time taken by the other body to reach the ground. Q. 15. A cylinder of mass
 M 
 and radius
r 
 is suspended at the corner of a room. Length of the thread is twice the radius of the cylinder. Find the tension in the thread and normal force applied by each wall on the cylinder assuming the walls to be smooth.
Thread
 Q. 16.
 
A rod of mass
 M 
 and length
 L
 lies on an incline having inclination of
q
= 37°. The coefficient of friction between the rod and the incline surface is
m 
 
= 0.90. Find the tension at the mid point of the rod. Q. 17.
 
A ball of mass
 M
is in equilibrium between a vertical wall and the inclined surface of a wedge. The inclination of the wedge is
q 
 = 45° and its mass is very small compared to that of the ball. The coefficient of friction between the wedge and the floor is
m 
 and there is no friction elsewhere. Find minimum value of
m 
 for which this equilibrium is possible.
q 
= 45°
M 
 Q.18
 
A helicopter of mass
 M 
 = 15000
kg
 is lifting a cubical box of mass
m
 = 2000
kg
. The helicopter is going up with an acceleration of
a
 = 1.2
m/s
2
. The four strings are tied at mid points of the sides of the square face
PQRS 
 of the box. The strings are identical and form a knot at
K 
. Another string
KH 
 connects the knot to the helicopter. Neglect mass of all strings and take g = 10
m/s
2
. Length of each string
 AK 
,
 BK 
,
CK 
 and
 DK 
 is equal to side length of the cube. (a) Find tension
T 
 in string
 AK 
. (b) Find tension
T 
0
 in string
KH.
 (c) Find the force (
F 
) applied by the atmosphere on the helicopter. Assume that the atmosphere exerts a negligible force on the box. (d) If the four strings are tied at
P,Q,R
 and
S 
 instead of
 A, B, C 
 &
 D
, how will the quantities
T 
,
T 
0
 and
F 
 change? Will they increase or decrease? Assume that length of the four identical strings remains same.
H K P AQ D C B  R S 
3
 
 3.4
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
 Q. 19. A pendulum has a bob connected to a light wire. Bob ‘
 A
’ is in equilibrium in the position shown. The string is horizontal and is connected to a block
 B
 resting on a rough surface. The block
 B
 is on verge of sliding when
q 
 = 60°.
q = 
 60°wireString
A B 
 (a) Is equilibrium possible if
q 
 were 70°? (b) With
q 
 
= 60°, calculate the ratio of tension in the pendulum wire immediately after the string is cut to the tension in the wire before the string is cut.
 
Q. 20. Two blocks of equal mass have been placed on two faces of a fixed wedge as shown in figure. The blocks are released from position where centre of one block is at a height
h
 above the centre of the other block. Find the time after which the centre of the two blocks will be at same horizontal level. There is no friction anywhere.
h 
60° 30°
 Q. 21. In the system shown in the figure, all surfaces are smooth. Block
 A
 and
 B
 have mass
m
 each and mass of block
C 
 is 2
m
. All pulleys are massless and fixed to block
C 
. Strings are light and the force
F 
 applied at the free end of the string is horizontal. Find the acceleration of all three blocks.
AB C F 
 Q. 22. A particle of mass
 M 
 rests on a rough inclined plane at an angle
q 
 to the horizontal
sin
q 
 =Ê Ë Á ˆ ¯ ˜ 
45
. It is connected to another mass
m
 as shown in fig. The pulley and string are light. The largest value of
m
 for which equilibrium is possible is
 M 
. Find the smallest value of
m
 for which equilibrium is possible.
m 
  M
 Q. 23. A small body
 A
 starts sliding down from the top of a wedge (see fig) whose base is equal to
󰁬
. The coefficient of friction between the body and wedge surface is
m
= 1.0. At what value of angle
q 
will the time of sliding be least?
A  
 Q. 24.
 
Three blocks
 A
,
 B
 and
C 
 each of mass
m
 are placed on a smooth horizontal table. There is no friction between the contact surfaces of the blocks as well. Horizontal force
F 
 is applied on each of
 A
 and
 B
 as shown. Find the ratio of normal force applied by the table on the three blocks (i.e.,
 R
 A
 :
 R
 B
 :
 R
C 
). Take
F  mg
=
23
F  AR 
A
30°
B  F R 
B 
30°
C R 
C 
 Q. 25. A
U 
 shaped container has uniform cross sectional area
S 
. It is suspended vertically with the help of a spring and two strings
 A
 and
 B
 as shown in the figure. The spring and strings are light. When water (density =
d 
 ) is poured slowly into the container it was observed that the level of water remained unchanged with respect to the ground. Find the force constant of the spring.
Spring
q  q 
AB 
3
 
 N
EWTON
’
S
 L
 AWS
 
3.5
 Q. 26. A uniform light spring has unstretched length of 3.0
m
. One of its end is fixed to a wall. A particle of mass
m
 = 20
g
 is glued to the spring at a point 1.0 m away from its fixed end. The free end of the spring is pulled away from the wall at a constant speed of 5
cm
 / 
s
. Assume that the spring remains horizontal (i.e., neglect gravity). Force constant of spring = 0.6 N /
cm
. (a) With what speed does the particle of mass
m
 move? (b) Find the force applied by the external agent pulling the spring at time 2.0 s after he started pulling.
5
cm / s 
3.0
m 
1.0
m m 
 
Q. 27. It was observed that a small block of mass
m
 remains in equilibrium at the centre of a vertical square frame, which was accelerated. The block is held by two identical light strings as shown. [Both strings are along the diagonal] (a) Which of 1, 2, 3 & 4 is/are possible direction/s of acceleration of the frame for block to remain in equilibrium inside it? (b) Find the acceleration of the frame for your answers to question (a).
1234
 
Q.28 In an emergency situation while driving one has tendency to jam the brakes, trying to stop in shortest distance. With wheels locked, the car slides and steering get useless. In ABS system the electronic sensors keep varying the brake pressure so as to keep the wheels rolling (without slipping) while ensuring that the friction remains limiting. Your friend has an old car with good brakes. He boasts saying that all the four wheels of his car get firmly locked and stop rotation immediately after the brakes are applied. You know that your new car which has a computerized anti lock braking system (
 ABS 
) is much safer. How will you convince your friend? In a typical situation, car without
 ABS 
 needs 20
m
 as minimum stopping distance. Under identical conditions, what minimum distance a car with
 ABS 
 would need to stop? Coefficient of kinetic friction between tyre and road is 25% less than the coefficient of static friction. Q. 29. Starting from rest a car takes at least ‘
t 
’ second to travel through a distance
s
 on a flat concrete road. Find the minimum time that will be needed for it to climb through a distance ‘
s
’ on an inclined concrete road. Assume that the car starts from rest and inclination of road is
q 
 
= 5° with horizontal. Coefficient of friction between tyres and the concrete road is
m 
 
= 1. Q. 30. A table cloth of length
 L
 is lying on a table with one of its end at the edge of the table. A block is kept at the centre of the table cloth. A man pulls the end of the table cloth horizontally so as to take it off the table. The cloth is pulled at a constant speed
V 
0
. What can you say about the coefficient of friction between the block and the cloth if the block remains on the table (i.e., it does not fall off the edge) as the cloth is pulled out. 
L— 
2
L— 
2
 Q. 31. A block rests on a horizontal surface. A horizontal force
F 
 is applied to the block. The acceleration (
a
) produced in the block as a function of applied force (
F 
) has been plotted in a graph (see figure). Find the mass of the block.
( )
ms 
–2
F (N) 
a
36 18
O 
 Q. 32. Repeat the last problem if the graph is as shown below.
( )
ms 
–2
36 18
a
F N 
( )
O 
5
 
 3.6
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
 Q. 33. A solid block of mass
m
 = 1
kg
 is resting on a horizontal platform as shown in figure. The
 z
 direction is vertically up. Coefficient of friction between the block and the platform is
m
= 0.2. The platform is moved with a time dependent velocity given by
V ti tj tk 
= + +
( )
23ˆˆˆm/s. Calculate the magnitude of the force exerted by the block on the platform. Take g = 10
m/s
2
z y x 
 Q. 34. In the system shown in the figure, the string is light and coefficient of friction between the 10
kg
 block and the incline surface is
m 
 
= 0.5. Mass of the hanger,
 H 
 is 0.5
kg
. A boy places a block of mass
m
 on the hanger and finds that the system does not move. What could be values of mass
m
? tan373410
2
∞ = =
and g
m/s
10
 kg H m 
37°
 Q. 35. A disc of mass
m
 lies flat on a smooth horizontal table. A light string runs halfway around it as shown in figure. One end of the string is attached to a particle of mass
m
 and the other end is being pulled with a force
F 
. There is no friction between the disc and the string. Find acceleration of the end of the string to which force is being applied.
m F m 
 Q. 36. (a) A car starts moving (at point
 A
) on a horizontal circular track and moves in anticlockwise sense. The speed of the car is made to increase uniformly. The car slips just after point
 D
. The figure shows the friction force (
 f 
) acting on the car at points
 A
,
 B
,
C 
 and
 D
. The length of the arrow indicates the magnitude of the friction and it is given that
 D >
 
 B
 >
C 
. At which point (
 A
,
 B
,
C 
 or
 D
) the friction forces represented is certainly wrong ?
C f D B AC f f D B f 
 (b)A particle is moving along an expanding spiral (shown in fig) such that the normal force on the particle [i.e., component of force perpendicular to the path of the particle] remains constant in magnitude. The possible direction of acceleration
a
( )
of the particle has been shown at three points A, B and C on its path. At which of these points the direction of acceleration has been represented correctly.
B
a
CVBAVA
a
VC
a
 (c) A particle is moving in XY plane with a velocity.
v i tj
= +
42
1
ˆˆ
–
ms. Calculate its rate of change of speed and normal acceleration at
t 
 = 2
s
. Q. 37. (i) A spinning disk has a hole at its centre. The surface of the disk is horizontal and a small block
 A
 of mass
m
 = 1
kg
 is placed on it.
w 
Ar B 
5
 
 N
EWTON
’
S
 L
 AWS
 
3.7
Block
 A
 is tied to a light inextensible string, other end of which passes through the hole and supports another block
 B
of mass
 M 
 = 2
kg
. The coefficient of friction between
 A
 and the disk surface is 0.5. It was observed that the disk is spinning with block
 A
 remaining at rest relative to the disk. Block
 B
 was found to be stationary. It was estimated that length of horizontal segment of the string (
r 
) was anywhere between 1.0
m
 to 1.5
m
. With this data what estimate can be made about the angular speed (
w 
) of the disk. [
g
= 10
m/s
2
] (ii) A spring has force constant equal to
k 
 = 100
 Nm
–1
. Ends of the spring are joined to give it a circular shape of radius
 R
 = 20
cm
. Now the spring is rotated about its symmetry axis (perpendicular to its plane) such that the circumference of the circle increases by 1%. Find the angular speed (
w 
). Mass of one meter length of the spring is
l
= 0.126
gm
–1
. Q. 38. Two particles of mass
m
1
 and
m
2
 are in space at separation
r 
 [vector from
m
1
 to
m
2
]. The only force that the two particles experience is the mutual gravitational pull. The force applied by
m
1
 on
m
2
is
F 
. Prove that
m 
 d r dt F 
22
=
 Where
m 
 m mm m
1212
+
 is known as reduced mass for the two particle system.
m 
1
r m 
2
F 
 Q. 39.
 
Six identical blocks – numbered 1 to 6 – have been glued in two groups of three each and have been suspended over a pulley as shown in fig. The pulley and string are massless and the system is in equilibrium. The block 1, 2, 3, and 4 get detached from the system in sequence starting with block 1. The time gap between separation of two consecutive block (i.e., time gap between separation of 1 and 2 or gap between separation of 2 and 3) is
t 
0
. Finally, blocks 5 and 6 remain connected to the string. (a) Find the final speed of blocks 5 and 6. (b) Plot the graph of variation of speed of block 5 with respect to time. Take
t
= 0 when block 1 gets detached.
642531
 Q. 40. Two monkeys
 A
 and
 B
 are holding on the two sides of a light string passing over a smooth pulley. Mass of the two monkeys are
m
 A
= 8
kg
 and
m
 B
 = 10
kg
 respectively [
g
 = 10
m/s
2
] (a) Monkey
 A
 holds the string tightly and
 B
 goes down with an acceleration
a
r 
 = 2
m/s
2
 relative to the string. Find the weight that
 A
 feels of his own body. (b) What is the weight experienced by two monkeys if
 A
 holds the string tightly and
 B
 goes down with an acceleration
a
r 
 = 4
m/s
2
 relative to the string.
A B 
L 
EVEL 
 
2
 Q. 41. Two strange particles
 A
 and
 B
 in space, exert no force on each other when they are at a separation greater than
 x 
0
 = 1.0
m
. When they are at a distance less than
 x 
0
, they repel one another along the line  joining them. The repulsion force is constant and does not depend on the distance between the particles. This repulsive force produces an acceleration of 6
ms
–2
 in
 A
 and 2
ms
–2
 in
 B
 when the particles are at separation less than
 x 
0
. In one experiment particle
 B
 is projected towards
 A
 with a velocity of 2
ms
–1
 from a large distance so as to
5
 
 3.8
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
hit
 A
 head on. The particle
 A
 is originally at rest and the system of two particles do not experience any external force. (a) Find the ratio of mass of
 A
 to that of
 B
. (b) Find the minimum distance between the particles during subsequent motion. (c) Find the final velocity of the two particles. Q. 42. A light string passing over a smooth pulley holds two identical buckets at its ends. Mass of each empty bucket is
 M 
 and each of them holds
 M 
 mass of sand. The system was in equilibrium when a small leak developed in bucket
 B
 (take this time as
t 
 = 0). The sand leaves the bucket at a constant rate of
m 
 
kg
 /s. Assume that the leaving sand particles have no relative speed with respect to the bucket (it means that there is no impulsive force on the bucket like leaving exhaust gases exert on a rocket). Find the speed (
V 
0
) of the two bucket when
 B
 is just empty.
A B 
 Q. 43. A chain is lying on a smooth table with half its length hanging over the edge of the table [fig(i)]. If the chain is released it slips off the table in time
t 
1
. Now, two identical small balls are attached to the two ends of the chain and the system is released [fig(ii)]. This time the chain took
t 
2
 time to slip off the table. Which time is larger,
t 
1
or
t 
2
?
(i) (ii)
 Q. 44. A triangular wedge
W 
 having mass
 M 
 is placed on an incline plane with its face
 AB
 horizontal. Inclination of the incline is
q 
. On the flat horizontal surface of the wedge there lies an infinite tower of rectangular blocks. Blocks 1, 2, 3, 4 ………. have masses
 M  M M M 
,,,248..................respectively. All surfaces are smooth. Find the contact force between the block 1 and 2 after the system is released from rest. Also find the acceleration of the wedge.
A B W 
1234
 Q. 45. In the system shown in fig, mass of the block is
m
1
= 4
kg
 and that of the hanging particle is
m
2
 = 1
kg
. The incline is fixed and surface is smooth. Block is initially held at the top of the incline and the particle hangs a distance
d 
 = 2.0 m below it. [Assume that the block and the particle are on same vertical line in this position]. System is released from this position. After what time will the distance between the block and the particle be minimum ? Find this minimum distance. [
g
 = 10
m/s
2
.]
d 
30°
 Q. 46.
 
A uniform chain of mass
 M 
 = 4.8
kg
 hangs in vertical plane as shown in the fig. (a) Show that horizontal component of tension is same throughout the chain. (b) Find tension in the chain at point
P
 where the chain makes an angle
q
= 15° with horizontal. (c) Find mass of segment
 AP
 of the chain. [Take
g
 = 10
m/s
2
; cos 15° = 0.96, sin 15° = 0.25]
P AB 
60°30°
q 
3
 
 N
EWTON
’
S
 L
 AWS
 
3.9
 Q. 47. Block
 A
 of mass
 M 
 is placed on an incline plane, connected to a string, passing over a pulley as shown in the fig. The other end of the string also carries a block
 B
 of mass
 M 
. The system is held in the position shown such that triangle
 APQ
 lies in a vertical plane with horizontal line
 AQ
 in the plane of the incline surface.
Q AM 
 a 
B M P 
q 
 Find the minimum coefficient of friction between the incline surface and block
 A
 such that the system remains at rest after it is released. Take
q 
=
a 
= 45°. Q. 48.Figure shown a fixed surface inclined at an angle
q 
 to the horizontal. A smooth groove is cut on the incline along
QR
 forming an angle
f 
 with
PR
. A small block is released at point
Q
 and it slides down to
 R
 in time
t 
. Find
t 
.
 
P  R Q d 
q 
 Q. 49. In the system shown in the figure
 AB
 and
CD
 are identical elastic cords having force constant
K 
. The string connected to the block of mass
 M 
 is inextensible and massless. The pulley is also massless. Initially, the cords are just taut. The end
 D
 of the cord
CD
 is gradually moved up. Find the vertical displacement of the end
 D
 by the time the block leaves the ground.
AB M C D 
 Q. 50. Blocks
 A
 and
 B
 have dimensions as shown in the fig. and their masses are 8
kg
 and 1
kg
 respectively.
 A
 small block
C 
 of mass 0.5
kg
 is placed on the top left corner of block
 A
. All surfaces are smooth. A horizontal force
F 
 = 18
 N 
 is applied to the block
 B
 at time
t 
 = 0. At what time will the block
C 
 hit the ground surface? Take
g
 = 10
m/s
2
. 
C AB F =N 
182.0
m 
4.0
m 
4.0
m 
 2.75
m 
 Q. 51. Three identical smooth balls are placed between two vertical walls as shown in fig. Mass of each ball is
m
 and radius is
r  R
=
59 where 2
 R
 is separation between the walls. (a) Force between which two contact surface is maximum? Find its value. (b) Force between which two contact surface is minimum and what is its value?
2R 
 Q. 52. A horizontal wooden block has a fixed rod
OA
 standing on it. From top point
 A
 of the rod, two wires have been fixed to points
 B
 and C on the block. The plane of triangle
OAB
 is perpendicular to the plane of the triangle
OAC 
. There are two identical beads on the two wires. One of the wires
O 
q q 
B C A
a
5
 
 3.10
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
is perfectly smooth while the other is rough. The wooden block is moved with a horizontal acceleration (
a
) that is perpendicular to the line
OB
 and it is observed that both the beads do not slide on the wire. Find the minimum coefficient of friction between the rough wire and the bead. Q. 53. In the arrangement shown in the fig. the pulley, the spring and the thread are ideal. The spring is stretched and the two blocks are in contact with a horizontal platform
P
. When the platform is gradually moved up by 2
cm
 the tension in the string becomes zero. If the platform is gradually moved down by 2
cm
 from its original position one of the blocks lose contact with the platform. Given
 M 
 = 4
kg
;
m
 = 2
kg
. (a) Find the force constant (
k 
) of the spring (b) If the platform continues to move down after one of the blocks loses contact, will the other block also lose contact? Assume that that the platform moves very slowly.
M  m P 
 Q. 54. In the arrangement shown in the fig. a monkey of mass
 M 
 keeps itself as well as block
 A
 at rest by firmly holding the rope. Rope is massless and the pulley is ideal. Height of the monkey and block
 A
 from the floor is
h
 and 2
h
 respectively [
h
 = 2.5
m
] (a) The monkey loosens its grip on the rope and slides down to the floor. At what height from the ground is block A at the instant the money hits the ground? (b) Another block of mass equal to that of
 A
 is stuck to the block
 A
 and the system is released. The monkey decides to keep itself at height
h
 above the ground and it allows the rope to slide through its hand. With what speed will the block strike the ground? (c) In the situation described in (b), the monkey decides to prevent the block from striking the floor. The monkey remains at height
h
 till the block crosses it. At the instant the block is crossing the monkey it begins climbing up the rope. Find the minimum acceleration of the monkey relative to the rope, so that the block is not able to hit the floor. Do you think that a monkey can climb with such an acceleration? (
g
 = 10
ms
 
–2
)
Ah h M 
Q. 55. An ideal spring is in its natural length (
 L
) with two objects
 A
 and
 B
 connected to its ends. A point
P
 on the unstretched spring is at a distance 23
 L
 from
 B
. Now the objects
 A
 and
 B
 are moved by 4
cm
 to the left and 8
cm
 to the right respectively. Find the displacement of point
P
.
LP 
23 L
8 cm 4 cm 
 Q. 56. The fig. shows an infinite tower of identical springs each having force constant
k 
. The connecting
F AB1
4
 
 N
EWTON
’
S
 L
 AWS
 
3.11
bars and all springs are massless. All springs are relaxed and the bottom row of springs is fixed to horizontal ground. The free end of the top spring is pulled up with a constant force
F 
. In equilibrium, find (a) The displacement of free end
 A
 of the top spring from relaxed position. (b) The displacement of the top bar
 B
1 from the initial relaxed position.
 
Q. 57. In the system shown in the fig. there is no friction and string is light. Mass of movable pulley
P
2
is
 M 
2
. If pulley
P
1
is massless, what should be value of applied force
F 
 to keep the system in equilibrium?
m 
P 
2
P 
1
F P 
3
 Q. 58. In the system shown in the fig., the bead of mass
m
 can slide on the string. There is friction between the bead and the string. Block has mass equal to twice that of the bead. The system is released from rest with length
l
of the string hanging below the bead. Calculate the distance moved by the block before the bead slips out of the thread. Assume the string and pulley to be massless.
P 
2
m 
2
 m bead 
 Q. 59.
 
In the arrangement shown in the fig. all pulleys are mass less and the strings are inextensible and light. Block
 A
 has mass
 M 
. (a) If the system stays at rest after it is released, find the mass of the block
 B
. (b) If mass of the block
 B
 is twice the value found in part (a) of the problem, calculate the acceleration of block
 A
.
M  AB 
 Q. 60. In the fig. shown, the pulley and string are mass less and the incline is frictionless. The segment
 AP
 of the string is parallel to the incline and the segment
PB
 is perpendicular to the incline. End of the string is pulled with a constant force
F 
. (a) If the block is moving up the incline with acceleration while being in contact with the incline, then angle
q 
must be less then
q 
0
. Find
q 
0
 (b) If
q  q 
=
0
2
 find the maximum acceleration with which the block can move up the plane without losing contact with the incline.
 P m
A
 F B
 Q.61. A triangular wedge
 A
 is held fixed and a block
 B
 is released on its inclined surface, from the top. Block
 B
 reaches the horizontal ground in time
t 
. In another experiment, the wedge
 A
 was free to slide on the horizontal surface and it took
t’
 time for the block
 B
 to reach the ground surface after it was released from the top. Neglect friction and assume
3
 
 3.12
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
that
 B
 remains in contact with
 A
. (a) Which time is larger
t 
 or
t 
´? Tell by simple observation. (b) When wedge
 A
 was free to move, it was observed that it moved leftward with an acceleration
g
4 and one of the two measured times (
t 
 &
t 
´) was twice the other. Find the inclination
q 
 of the inclined surface of the wedge.
B A
 Q. 62.A block
 A
 is made to move up an inclined plane of inclination
q 
 with constant acceleration
a
0
 as shown in figure. Bob
 B
, hanging from block
 A
by a light inextensible string, is held vertical and is moving along with the block. Calculate the magnitude of acceleration of block
 A
 relative to the bob immediately after bob is released.
a
0
AB 
 Q. 63.
 
A 50
kg
 man is standing at the centre of a 30
kg
 platform
 A
. Length of the platform is 10 m and coefficient of friction between the platform and the horizontal ground is 0.2. Man is holding one end of a light rope which is connected to a 50
kg
 box
 B
. The coefficient of friction between the box and the ground is 0.5. The man pulls the rope so as to slowly move the box and ensuring that he himself does not move relative to the ground. If the shoes of the man does not slip on the platform, calculate how much time it will take for the man to fall off the platform. Assume that rope remains horizontal, and coefficient of friction between shoes and the platform is 0.6. Q. 64. A wedge is placed on the smooth surface of a fixed incline having inclination
q 
 with the horizontal. The vertical wall of the wedge has height
h
 and there is a small block
 A
on the edge of the horizontal surface of the wedge. Mass of the wedge and the small block are
 M 
 and
m
 respectively. (a) Find the acceleration of the wedge if friction between block
 A
 and the wedge is large enough to prevent slipping between the two. (b) Find friction force between the block and the wedge in the above case. Also find the normal force between the two. (c) Assuming there is no friction between the block and the wedge, calculate the time in which the block will hit the incline.
q q 
h A
 Q. 65.
 
In the system shown in figure, all surfaces are smooth, pulley and strings are massless. Mass of both
 A
 and
 B
 are equal. The system is released from rest.
AB
 (a) Find the
a a
 A B
.
 immediately after the system is released.
a
 A
 and
a
 B
 are accelerations of block
 A
 and
 B
 respectively. (b) Find
a
 A
 immediately after the system is released. Q.66. A block is placed on an incline having inclination
q 
. There is a rigid
 L
 shaped frame fixed to the block. A plumb line (a ball connected to a thread) is attached to the end
 A
 of the frame. The system is released on the inline. Find the angle
A 30
 kg 
50
 kg 
B10
 m 
4
 
 N
EWTON
’
S
 L
 AWS
 
3.13
that the plumb line will make with vertical in its equilibrium position relative to the block when (a) the incline is smooth (b) there is friction and the acceleration of the block is half its value when the incline is smooth
AB 
q 
 Q. 67.
 
A wedge of mass
m
 is placed on a horizontal smooth table. A block of mass
m
 is placed at the mid point of the smooth inclined surface having length
 L
 along its line of greatest slope. Inclination of the inclined surface is
q 
= 45°. The block is released and simultaneously a constant horizontal force
F 
 is applied on the wedge as shown. (a) What is value of
F 
 if the block does not slide on the wedge? (b) In how much time the block will come out of the incline surface if applied force is 1.5 times that found in part (a)
q 
Lm F m 
 Q. 68. A rod is kept inclined at an angle
q 
 with the horizontal
 A
 sleeve of mass
m
 can slide on the rod. If the coefficient of friction between the rod and the sleeve is
m 
, for what values of horizontal acceleration
a
 of the rod, towards left, the sleeve will not slide over the rod?
q 
a
 Q. 69. In the arrangement shown in figure, a block
 A
 of mass
m
 has been placed on a smooth wedge
 B
 of mass
 M 
. The wedge lies on a horizontal smooth surface. Another block
C 
 of mass
 M 
4 has been placed in contact with the wedge
 B
 as shown. The coefficient of friction between the block
C 
 and the vertical wedge wall is
m 
 =
34
. Find the ratio
m M 
 for which the block
C 
 will not slide with respect to the wedge after the system is released?
Am C M 4 M B 30° 
 Q. 70. A smooth rod is fixed at an angle
a 
to the horizontal. A small ring of mass
m
 can slide along the rod. A thread carrying a small sphere of mass
 M 
 is attached to the ring. To keep the system in equilibrium, another thread is attached to the ring which carries a load of mass
m
0
 at its end (see figure). The thread runs parallel to the rod between the ring and the pulley. All threads and pulley are massless. (a) Find
m
0
 so that system is in equilibrium. (b) Find acceleration of the sphere
 M 
 immediately after the thread supporting
m
0
 is cut.
m m 
0
M 
a 
 Q. 71. In the system shown in figure all surfaces are smooth and string and pulleys are light. Angle of wedge
q 
 = Ê Ë Á ˆ ¯ ˜ 
-
sin
1
35. When released from rest it was found that the wedge of mass
m
0
 does not move. Find
 M m
.
m 
0
m M 
q 
 Q. 72. In the last problem take
 M
=
m
 and
m
0
 = 2
m
 and calculate the acceleration of the wedge.
 
Q. 73. In the system shown in the figure all surfaces are smooth, pulley and string are massless. The string between the two pulleys and between pulley and
4
 
 3.14
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
block of mass 5 m is parallel to the incline surface of the block of mass 4 m. The system is released from rest. Find the acceleration of the block of mass 4 m. tan3734
∞ =ÈÎ͢˚˙
5
m 
4
m 
37°8
m 
 Q. 74. In the system shown in figure, the two springs
S 
1
 and
S 
2
 have force constant
k 
 each. Pulley, springs and strings are all massless. Initially, the system is in equilibrium with spring
S 
1
 stretched and
S 
2
 relaxed. The end
 A
 of the string is pulled down slowly through a distance
 L
. By what distance does the block of mass
 M 
 move?
S 
2
S 
1
M A
 Q. 75. The system shown in figure is in equilibrium. Pulley, springs and the strings are massless. The three blocks
 A
,
 B
 and
C 
 have equal masses.
 x 
1
 and
 x 
2
 are extensions in the spring 1 and spring 2 respectively.
Spring 1Spring 2
AB 
 (a) Find the value of
d x dt 
222
 immediately after spring 1 is cut. (b) Find the value of
d x dt 
212
 and
d x dt 
222
 immediately after string
 AB
 is cut. (c) Find the value of
d x dt 
212
 and
d x dt 
222
 immediately after spring 2 is cut. Q. 76. In the figure shown, the pulley, strings and springs are mass less. The block is moved to right by a distance
 x 
0
 from the position where the two springs are relaxed. The block is released from this position.
Smooth
K 
1
K 
2
T 
0
M 
 (a) Find the acceleration of the block immediately after it is released. (b) Find tension (
T 
0
) in the support holding the pulley to the wall, immediately after the block is released. Assume no friction. Q.77. The system shown in figure is in equilibrium. Surface
PQ
 of wedge
 A
, having mass
 M 
, is horizontal. Block
 B
, having mass 2
 M 
, rests on wedge
 A
 and is supported by a vertical spring. The spring balance
S 
 is showing a reading of 2
 Mg
. There is no friction anywhere and the thread
QS 
 is parallel to the incline surface. The thread
QS 
 is cut. Find the acceleration of
 A
 and the normal contact force between
 A
 and
 B
 immediately after the thread is cut.
P M 
q 
 = 
 45°
2M B AQ S 
 Q.78. A triangular wedge of mass
 M 
 lies on a smooth horizontal table with half of its base projecting out of the edge of the table. A block of mass
m
 is kept at the top of the smooth incline surface of the
3
 
 N
EWTON
’
S
 L
 AWS
 
3.15
wedge and the system is let go. Find the maximum value of
 M m
 for which the block will land on the table. Take
q 
= 60°.
M m 
q 
L
2
L
2
 Q.79.
 
In the system shown in the figure all surfaces are smooth and both the pulleys are mass less. Block on the incline surface of wedge
 A
 has mass
m
. Mass of
 A
 and
 B
 are
 M 
 = 4
m
 and
 M 
0
 = 2
m
 respectively. Find the acceleration of wedge
 A
when the system is released from rest.
m AB 
q 
 Q.80. A block of mass
m
 requires a horizontal force
F 
0
 to move it on a horizontal metal plate with constant velocity. The metal plate is folded to make it a right angled horizontal trough. Find the horizontal force
F 
 that is needed to move the block with constant velocity along this trough.
F 
0
F 
 45°
 Q.81. Block
 A
 of mass m
A
 = 200
g
 is placed on an incline plane and
a
 constant force
F 
 = 2.2
 N 
 is applied on it parallel to the incline. Taking the initial position of the block as origin and up along the incline as
 x 
 direction, the position (
 x 
) time (
t 
) graph of the block is recorded (see figure (b)). The same experiment is repeated with another block
 B
 of mass m
B
 = 500
g
. Same force
F 
 is applied to it up along the incline and its position – time graph is recorded (see figure (b)). Now the two blocks are connected by a light string and released on the same incline as shown in figure (c). Find the tension in the string. tan;/ 
q 
 = =ÈÎ͢˚˙
3410
2
g m s
q q 
F x 
 = 0
x(m) 
2.00.5–0.5–2.0
AA
1.0
 
 2.0
 t 
 (sec)
 AB O x 
 Q.82. Block
 B
 of mass
m
 has been placed on block
 A
 of mass 3
m
 as shown. Block
 A
 rests on a smooth horizontal table.
F 
1
 is the maximum horizontal force that can be applied on the block
 A
 such that there is no slipping between the blocks. Similarly,
F 
2
 is the maximum horizontal force that can be applied on the block
 B
 so that the two blocks move together without slipping on each other. When
F 
1
 and
F 
2
 both are applied together as shown in figure.
Smooth
AB 
3
 m F 
2
F 
1
m 
 (a) Find the friction force acting between the blocks. (b) Acceleration of the two blocks. (c) If
F 
2
 is decreased a little, what will be direction of friction acting on
 B
.Q. 83. (i) In the arrangement shown in the figure the coefficient of friction between the 2
kg
 block and the vertical wall is
m 
= 0.5. A constant horizontal force of 40 N keeps the block pressed against the wall. The spring has a natural length of 1.0 m and its force constant is
k 
 = 400
 Nm
–1
. What should be the height
h
 of the block above the horizontal floor for it to be in equilibrium. The spring is not tied to the block.
F =N 
402
kg h 
 (ii) A block of mass
 M 
 is pressed against a rough vertical wall by applying a force
F 
 making an
3
 
 3.16
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
angle of
q 
 with horizontal (as shown in figure). Coefficient of friction between the wall and the block is
m 
 = 0.75.
 M  F 
q m 
 (a) If
F 
 = 2
 Mg
, find the range of values of
q 
 so that the block does not slide [Take tan 37° = 0.75; sin 24° = 0.4] (b) Find the maximum value of
q 
 above which equilibrium is not possible for any magnitude of force
F 
. Q. 84. A block is projected up along a rough incline with a velocity of
u 
= 10
m/s
. After 4
s
 the block was at point
 B
 at a distance of 5
m
 from the starting point
 A
 and was travelling down at a velocity of
v
 = 4
m/s
.
AB 
 v = 4 m / s u = m / s
  1  0  5
 m
q 
 (a) Find time after projection at which the block came to rest. (b) Find the coefficient of friction between the block and the incline. Take
g
= 10
m/s
2
 Q. 85. A long piece of paper is being pulled on a horizontal surface with a constant velocity
u 
 along its length. Width of the paper is
 L
. A small block moving horizontally, perpendicular to the direction of motion of the paper, with velocity
v
 slides onto the paper. The coefficient of friction between the block and the paper is
m 
. Find maximum value of
v
 such that the block does not cross the opposite edge of the paper.
u Lv 
 
Q. 86. A block of mass
m
 = 1
kg
 is kept pressed against a spring on a rough horizontal surface. The spring is compressed by 10
cm
 from its natural length and to keep the block at rest in this position a horizontal force (
F 
) towards left is applied. It was found that the block can be kept at rest if 8
 N 
 
F 
 18
 N 
. Find the spring constant (
k 
) and the coefficient of friction (
m 
) between the block and the horizontal surface.
m F 
 Q. 87. An experimenter is inside a uniformly accelerated train. Train is moving horizontally with constant acceleration
a
0
. He places a wooden plank
 AB
 in horizontal position with end
 A
 pointing towards the engine of the train. A block is released at end
 A
 of the plank and it reaches end
 B
 in time
t 
1
. The same plank is placed at an inclination of 45° to the horizontal. When the block is released at
 A
 it now climbs to
 B
 in time
t 
2
. It was found that
t t 
21
2
54
=
. What is the coefficient of friction between the block and the plank?
Direction of acceleration of the train
B  AB 
45
A
 Q. 88. Two hemispheres of radii
 R
 and
r 
 (<
 R
) are fixed on a horizontal table touching each other (see figure). A uniform rod rests on two spheres as shown. The coefficient of friction between the rod and two spheres is
 µ
. Find the minimum value of the ratio 
r  R
 for which the rod will not slide.
O 
1
 O 
2
r R 
 Q. 89. In order to lift a heavy block
 A
, an engineer has designed a wedge system as shown. Wedge
C 
 is fixed. A horizontal force
F 
 is applied to
 B
 to lift block
 A
. Wedge
 B
 itself has negligible mass and mass of
 A
 is
 M 
. The coefficient of friction at all
4
 
 N
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3.17
surfaces is
m 
. Find the value of applied force
F 
 at which the block
 A
 just begins to rise.
q q 
C 
 (fixed)
B A
 Q.90. A 60
kg
 platform has been placed on a rough incline having inclination
q 
= 37°. The coefficient of friction between the platform and the incline is
m 
= 0.5. A 40
kg
 man is running down on the platform so as to keep the platform stationary. What is the acceleration of the man? It is known that the man cannot manage to go beyond an acceleration of 7
m/s
2
.
sin3735
∞ =ÈÎ͢˚˙
 
q 
  
 m
 
 0.  5
40
 kg 
60
 kg 
 Q. 91. In the system show in figure, mass of the block placed on horizontal surface is
 M 
 = 4
kg
. A constant horizontal force of
F 
 = 40
 N 
 is applied on it as shown. The coefficient of friction between the blocks and surfaces is
m 
= 0.5. Calculate the values of mass
m
 of the block on the incline for which the system does not move. sin;/ 373510
2
∞ = =ÈÎ͢˚˙
g m s
q 
 =
 37°
m M F 
 Q. 92. In the arrangement shown in the figure, block
 A
 of mass 8
kg
 rests on a horizontal table having coefficient of friction
m 
= 0.5. Block
 B
 has a mass of 6
kg
 and rests on a smooth incline having inclination angle
q 
 = Ê Ë Á ˆ ¯ ˜ 
-
sin
1
25
. All pulleys and strings are mass less. Mass of block
C 
 is
 M 
. [
g
= 10
m/s
2
]
q 
AC B 
 (a) Find value of
 M 
 for which block
 B
 does not accelerate (b) Find maximum value of
 M 
 for which
 A
 does not accelerate. Q.93. In the arrangement shown in figure, pulley and string are light. Friction coefficient between the two blocks is
m 
 whereas the incline is smooth. Block
 A
 has mass
m
 and difference in mass of the two blocks is
m
. Find minimum value of
m 
 for which the system will not accelerate when released from rest.
AB 
 Q. 94. In the arrangement shown in figure pulley
P
 can move whereas other two pulleys are fixed. All of them are light. String is light and inextensible. The coefficient of friction between 2
kg
 and 3
kg
 block is
m 
= 0.75 and that between 3
kg
 block and the table is
m 
= 0.5. The system is released from rest
2
 kg 
3
 kg P M 
 (i) Find maximum value of mass
 M 
, so that the system does not move. Find friction force between 2
kg
 and 3
kg
 blocks in this case. (ii) If
 M 
 = 4
kg
, find the tension in the string attached to 2
kg
 block.
3
 
 3.18
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
 (iii) If
 M 
 = 4
kg
 and
m 
1
= 0.9, find friction force between the two blocks, and acceleration of
 M 
. (iv) Find acceleration of
 M 
 if
m 
1
= 0.75,
m 
2
= -0.9 and
 M 
 = 4
kg
. Q. 95. A rope of length
p
21
+Ê Ë Á ˆ ¯ ˜ 
 
 R
 has been placed on a smooth sphere of radius
 R
 as shown in figure. End
 A
 of the rope is at the top of the sphere and end
 B
 is overhanging. Mass per unit length of the rope is . The horizontal string holding this rope in place can tolerate tension equal to weight of the rope. Find the maximum mass (
 M 
0
) of a block that can be tied to the end
 B
 of the rope so that the string does not break.
StringRope
AR B 
 Q.96. A uniform rope has been placed on a sloping surface as shown in the figure. The vertical separation and horizontal separation between the end points of the rope are
 H 
 and
 X 
 respectively. The friction coefficient (
m 
) is just good enough to prevent the rope from sliding down. Find the value of
m 
.
AB H X 
 Q.97. A uniform rope
 ABCDE 
 has mass
 M 
 and it is laid along two incline surfaces (
 AB
 and
CD
) and two horizontal surfaces (
 BC 
 and
 DE 
) as shown in figure. The four parts of the rope
 AB
,
 BC 
,
CD
 and
 DE 
 are of equal lengths. The coefficient of friction (
m 
) is uniform along the entire surface and is just good enough to prevent the rope from sliding.
Vertically down
AB  C D  E 
60°30°
 (a) Find
m 
 (b)
 x 
 is distance measured along the length of the rope starting from point
 A
. Plot the variation of tension in the rope (
T 
) with distance
 x 
. (c) Find the maximum tension in the rope. Q. 98. (i) Four small blocks are interconnected with light strings and placed over a fixed sphere as shown. Blocks
 A
,
 B
 and
C 
 are identical each having mass
m
 = 1
kg
. Block
 D
 has a mass of
m´ 
 = 2
kg
. The coefficient of friction between the blocks and the sphere is
m 
= 0.5. The system is released from the position shown in figure.
D AB C 
37°53°37°
 (a) Find the tension in each string. Which string has largest tension? (b) Find the friction force acting on each block. 
Taketan;/ 373410
2
∞ = =ÈÎ͢˚˙
g m s
 (ii) A fixed square prism
 ABCD
 has its axis horizontal and perpendicular to the plane of the figure. The face
 AB
 makes 45° with the vertical. On the upper faces
 AB
 and
 BC 
 of the prism there are light bodies
P
 and
Q
 respectively. The two bodies (
P
 and
Q
) are connected using a string
S 
1
 and strings
S 
0
 and
S 
2
 are hanging from
P
 and
Q
 respectively. All strings are mass less, and inextensible. String
S 
1
 is horizontal and the other two strings are vertical. The coefficient of friction between the bodies and the prism is
m 
. Assume that
P
 and
Q
 always remain in contact with the prism.
3
 
 N
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3.19
S 
2
S 
0
S 
1
AB C D P  Q 
45°
 (a) If tension in
S 
0
 is
T 
0
, find the minimum tension (
T 
1
) in
S 
1
 to keep the body
P
 at rest. (b) A mass
 M 
0
 is tied to the lower end of string
S 
0
 and another mass
m
2
 is tied to
S 
2
. Find the minimum value of
m
2
 so as to keep
P
 and
Q
 at rest.
 
Q. 99. A metal disc of radius
 R
 can rotate about the vertical axis passing through its centre. The top surface of the disc is uniformly covered with dust particles. The disc is rotated with gradually increasing speed. At what value of the angular speed (
w 
) of the disc the 75% of the top surface will become dust free. Assume that the coefficient of friction between the dust particles and the metal disc is
m 
= 0.5. Assume no interaction amongst the dust particles.
R  
 Q.100. In the last question, the axis of the disc is tilted slightly to make an angle
q 
 with the vertical. Redo the problem for this condition and check the result by putting
q 
= 0 in your answer. Q. 101.
 A sphere of mass
 M 
 is held at rest on a horizontal floor. One end of a light string is fixed at a point that is vertically above the centre of the sphere. The other end of the string is connected to a small particle of mass
m
 that rests on the sphere. The string makes an angle
a 
= 30° with the vertical. Find the acceleration of the sphere immediately after it is released. There is no friction anywhere.Q. 102. A light rod
 AB
 is fitted with a small sleeve of mass
m
 which can slide smoothly over it. The sleeve is connected to the two ends of the rod using two springs of force constant 2
k 
 and
k 
 (see fig). The ends of the springs at
 A
 and
 B
 are fixed and the other ends (connected to sleeve) can move along with the sleeve. The natural length of spring connected to
 A
 is
󰁬
0
. Now the rod is rotated with angular velocity
w 
 about an axis passing through end A that is perpendicular to the rod. Take
k m
w h
2
 =
 and express the change in length of each spring (in equilibrium position of the sleeve relative to the rod) in terms of
󰁬
0
and
h
.
A
w 
2K m KB
 Q. 103. A metallic hemisphere is having dust on its surface. The sphere is rotated about a vertical axis passing through its centre at angular speed
w 
 = 10
rad 
 
s
 
–1
. Now the dust is visible only on top 20% area of the curved hemispherical surface. Radius of the hemisphere is
 R
 = 0.1
m
. Find the coefficient of friction between the dust particle and the hemisphere [
g
 = 10
ms
 
–2
].
       w
Dust
 Q. 104. Civil engineers bank a road to help a car negotiate a curve. While designing a road they usually ignore friction. However, a young engineer decided to include friction in his calculation while designing a road. The radius of curvature of the road is
 R
 and the coefficient of friction between the tire and the road is
m 
. (a) What should be the banking angle (
q 
) so that car travelling up to a maximum speed
V 
0
can negotiate the curve.
3
 
 3.20
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
 (b) At what speed (
V 
1
) shall a car travel on a road banked at
q 
0
so that there is no tendency to skid. (No tendency to skid means there is no static friction force action on the car). (c) The driver of a car travelling at speed (
V 
1
) starts retarding (by applying brakes). What angle (acute, obtuse or right angle) does the resultant friction force on the car make with the direction of motion? Q. 105.
 
A turn of radius 100
m
 is banked for a speed of 20
m/s
 (a) Find the banking angle (b) If a vehicle of mass 500
kg
 negotiates the curve find the force of friction on it if its speed is – (i) 30
m/s
 (ii) 10
m/s
 Assume that friction is sufficient to prevent skidding and slipping. [Take tan 22° = 0.4, sin 22° = 0.375, cos 22° = 0.93,
g
 = 10
ms
–2
] Q. 106. A horizontal circular turning has a curved length
 L
 and radius
 R
. A car enters the turn with a speed
V 
0
 and its speed increases at a constant rate
 f 
. If the coefficient of friction is
m 
, (a) At what time
t 
0
, after entering the curve, will the car skid? (Take it for granted that it skids somewhere on the turning) (b) At a time
t 
 (<
t 
0
) what is the force of friction acting on the car? Q. 107. A 70
kg
 man enters a lift and stands an a weighing scale inside it. At time
t 
 = 0, the lift starts moving up and stops at a higher floor at
t 
 = 9.0
s
. During the course of this journey, the weighing scale records his weight and given a plot of his weight vs time. The plot is shown in the fig. [Take
g
 = 10
m/s
2
] (a) Find
F 
0
 (b) Find the magnitude of maximum acceleration of the lift. (c) Find maximum speed acquired by the lift.
800
N F 
700
N 
o0.2 2.8 3.06.0 9.0
t 
 (s)
Fo 
 Q.108.Three small discs are connected with two identical massless rods as shown in fig. The rods are pinned to the discs such that angle between them can change freely. The system is placed on a smooth horizontal surface with discs
 A
 and
 B
 touching a smooth wall and the angle
 ACB
 being 90°. A force
F 
 is applied to the disc C in a direction perpendicular to the wall. Find acceleration of disc
 B
 immediately after the force starts to act. Masses of discs are
m
 A
 =
m
;
m
 B
= 2
m
;
m
C
=
m
 [wall is perpendicular to the plane of the fig.]
90°wall2mBmCFAm
 Q. 109. Figure shows two blocks in contact placed on an incline of angle
q 
= 30°. The coefficient of friction between the block of mass 4
kg
 and the incline is
m 
1
, and that between 2
kg
 block and incline is
m 
2
. Find the acceleration of the blocks and the contact force between them if – (a)
m 
1
= 0.5,
m 
2
= 0.8 (b)
m 
1
= 0.8,
m 
2
= 0.5 (c)
m 
1
= 0.6,
m 
2
= 0.1 
[Take
g
 = 10
m/s
2
]
= 30°
 4   k g 2   k g
 Q. 110. A small collar of mass
m
 = 100
g
 slides over the surface of a horizontal circular rod of radius
 R
 = 0.3
m
. The coefficient of friction between the rod and the collar is
m 
 
= 0.8. Find the angle made with vertical by the force applied by the rod on the collar when speed of the collar is
V 
 = 2
m/s
.
4
 
 N
EWTON
’
S
 L
 AWS
 
3.21
 Q. 111.A flat race track consists of two straight section
 AC 
 and
 DB
 each of length 180
m
 and one semi circular section
 DC 
 of radius
 R
 = 150
m
. A car starting from rest at
 A
 has to reach
 B
 in least possible time (the car may cross through point
 B
 and need not stop there). The coefficient of friction between the tyres and the road is
m 
 
= 0.6 and the top speed that the car can acquire is 180
kph
. Find the minimum time needed to move from A to B under ideal conditions. Braking is not allowed in the entire journey [
g
 = 10
m/s
 
2
]
180 m180 mROBDCA
 Q. 112. A small insect is climbing slowly along the inner wall of a hemispherical bowl of radius R. The insect is unable to climb beyond
q 
 = 45°. Whenever it tries to climb beyond
q 
 = 45°, it slips. (a) Find the minimum angular speed
w 
 with which the bowl shall be rotated about its vertical radius so that the insect can climb upto
q 
 = 60°. (b) Find minimum
w 
 for which the insect can move out of the bowl.
R
w 
O
 Q. 113.
 C D B A
 A room is in shape of a cube.
 A
 heavy ball (
 B
) is suspended at the centre of the room tied to three inextensible strings as shown. String
 BA
 is horizontal with A being the centre point of the wall. Find the ratio of tension in the string
 BA
and
 BC 
.Q.114. Two identical smooth disc of radius
 R
 have been placed on a frictionless table touching each other. Another circular plate is placed between them as shown in figure. The mass per unit area of each object is
s 
, and the line joining the centers of the plate and the disc is
q 
 (a) Find the minimum horizontal force F
0
 that must be applied to the two discs to keep them together. (b) Angle
q 
 can be changed by changing the size of the circular plate. Find
F 
0
 when
q 
0. 
useandforsmallcossin
q  q q q q 
= - =ÈÎ͢˚˙
12
2
 (c) Find
F 
0
 when
q  p 
Æ
2
. Explain the result.
F F  R 
 
Q. 115. Three identical smooth cylinders, each of mass
m
 and radius
r 
 are resting in equilibrium within a fixed smooth cylinder of radius
 R
 (only a part of this cylinder has been shown in the fig). Find the largest value of
 R
 in terms of
r 
 for the small cylinders to remain in equilibrium.Q. 116. A massless spring of force constant
K 
 and natural length
󰁬
0
 is hanging from a ceiling. An insect of mass
m
 is sitting at the lower end of the spring and the system is in equilibrium . The insect starts slowing climbing up the spring so as to eat a bug sitting on the ceiling. Assume that insect climbs
3
 
 3.22
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
 without slipping on the spring and
K  mg
󰁬
0
. Find the length of the spring when the insect is at
4th
 of its original distance from the bug.
KBuginsect
 
Q. 117. In the system shown in fig., all pulleys are mass less and the string is inextensible and light. (a) After the system is released, find the acceleration of mass
m
1
 (b) If
m
1
 = 1
kg
,
m
2
 = 2
kg
 and
m
3
= 3
kg
 then what must be value of mass
m
4
 so that it accelerates downwards?
m 
1
 m 
2
 m 
3
m 
4
Q. 118. In the system shown in fig., block
 A
 and
C 
 are placed on smooth floors and both have mass equal to
m
1
. Blocks
 B
 and
 D
 are identical having mass
m
2
 each. Coefficient of friction
 
BAFDC
 Between
 A
 and
 B
 and that between
C 
 and
 D
 are both equal to
m 
. String and pulleys are light.
 A
 horizontal force F is applied on block
C 
 and is gradually increased. (a) Find the maximum value of
F 
 (call it
F 
0
) so that all the four blocks move with same acceleration. (b) Will the value of
F 
0
 increase or decrease if another block (
 E 
) of mass
m
2
 is placed above block
 D
 and coefficient of friction between
 E 
 and
 D
 is
m 
?
 
Q. 119. A chain with uniform mass per unit length lies in a vertical plane along the slope of a smooth hill. The two end of the chain are at same height. If the chain is released from this position find its acceleration.Q. 120. A uniform rope of length
p 
 R
2
 has been placed on fixed cylinder of radius
 R
 as shown in the fig. One end of the rope is at the top of the cylinder. The coefficient of friction between the rope and the cylinder is just enough to prevent the rope from sliding. Mass of the rope is
 M 
. (a) At what position, the tension in the rope is maximum? (b) Calculate the value of maximum tension in the rope.
R
 
Q. 121. In the last problem, the rope is placed on the cylinder as shown. Find maximum tension in the rope.
45° 45°
Q. 122. A 4
kg
 block is placed on a rough horizontal surface. The coefficient of friction between the
4
 
 N
EWTON
’
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 AWS
 
3.23
block and the surface is
m 
= 0.5. A force
F 
 = 18
 N 
 is applied on the block making an angle
q 
 with the horizontal. Find the range of values of
q 
 for which the block can start moving. 
Take = 10 m/s
g
211
26310912584,tan()sin.
--
= ∞Ê Ë Áˆ ¯ ˜  = ∞ÈÎÍÍ͢˚˙˙˙˙
4kg
F = 18 N
 
m 
= 0.5
Q. 123. Two rectangular blocks
 A
 and
 B
 are placed on a horizontal surface at a very small separation. The masses of the blocks are
m
 A
 = 4
kg
 and
m
 B
 = 5
kg
. Coefficient of friction between the horizontal surface and both the blocks is
m 
= 0.4. Horizontal forces
F 
1
and
F 
2
 are applied on the blocks as shown. Both the forces vary with time as 
F 
1
 = 15 + 0.5
t 
 
F 
2
 = 2
t 
 Where ‘
t 
’ is time in second. Plot the variation of friction force acting on the two blocks (
 A
 
and
 B
) vs time till the motion starts. Take rightward direction to be positive for
 B
 and leftward direction to be positive for
 A
.
A B
F
1
4 kg5 kgF
2
L 
EVEL 
 3
 
Q. 124. A rope of mass
m
 is hung from a ceiling. The centre point is pulled down with a vertical force
F 
. The tangent to the rope at its ends makes an angle
a 
 with horizontal ceiling. The two tangents at the lower point make an angle of
q 
 with each other. Find
q 
.
q a  a 
F  
 
Q. 125. A smooth cylinder is fixed with its axis horizontal. Radius of the cylinder is
 R
. A uniform rope (
 ACB
) of linear mass density (
kg
 /m) is exactly of length
 R
 and is held in semicircular shape in vertical plane around the cylinder as shown in figure. Two massless strings are connected at the two ends of the rope and are pulled up vertically with force
T 
0
 to keep the rope in contact with the cylinder. (a) Find minimum value of
T 
0
 so that the rope does not lose contact with the cylinder at any point. (b) If
T 
0
 is decreased slightly below the minimum value calculated in (a), where will the rope lose contact with the cylinder.
T 
0
 T 
0
B O R C A
 
Q. 126. A block of mass
m
 placed on an incline just begins to slide when inclination of the incline is made
q 
= 45°. With inclination equal to
q 
= 30°, the block is placed on the incline. A horizontal force (
F 
) parallel to the surface of the incline is applied to the block. The force
F 
 is gradually increased from zero. At what angle
a 
 to the force
F 
 will the block first begin to slide?
F  
q a 
 
Q. 127. In the last problem if it is allowed to apply the force
F 
 in any direction, find the minimum force
F 
min
 needed to move the block on the incline.
 
Q. 128. A block
 A
 has been placed symmetrically over two identical blocks
 B
 and
C 
. All the three blocks have equal mass,
 M 
 each, and the horizontal surface on which
 B
 and
C 
 are placed is smooth. The coefficient of friction between
 A
 and either of
 B
 and
C 
 is
m 
. The block
 A
 exerts equal pressure on
 B
 and
C 
. A horizontal force
F 
 is applied to the
3
 
 3.24
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
block
 B
.
F B C A
 (a) Find maximum value of
F 
 so that
 A
 does not slip on
 B
 or
C 
 and the three blocks move together. (b) If
F 
 is increased beyond the maximum found in (a) where will we see slipping first- at contact of
 A
 and
 B
 or at the contact of
 A
 and
C 
. (c) If
F 
 is kept half the maximum found in (a), calculate the ratio of friction force between
 A
 and
 B
 to that between
 A
 and
C 
. Does this ratio change if
F 
 is decreased further?
 
Q. 129. In the arrangement shown in the figure the coefficient of friction between the blocks
C 
 and
 D
 is
m 
= 0.7 and that between block
 D
 and the horizontal table is
m 
= 0.2. The system is released from rest. [ Take g = 10 ms
–2
] Pulleys and threads are massless.
C D B 
1.5
 kg 
3
 kg 
3
 kg 
0.5
 kg 
A
 (a) Find the acceleration of the block
C 
. (b) Block
 B
 is replaced with a new block. What shall be the minimum mass of this new block so that block
C 
 and
 D
 accelerate in opposite direction?
 
Q. 130. A hemisphere of mass
 M 
 and radius
 R
 rests on a smooth horizontal table. A vertical rod of mass
m
 is held between two smooth guide walls supported on the sphere as shown. There is no friction between the rod and the sphere. A horizontal string tied to the sphere keeps the system at rest.
String
M  R m 
q 
 (a) Find tension in the string. (b) Find the acceleration of the hemisphere immediately after the string is cut.
 
Q. 131. A semicircular ring of radius
 R
 is fixed on a smooth horizontal table. A small block is projected with speed
u
 so as to enter the ring at end
 A
. Initial velocity of the block is along tangent to the ring at
 A
 and it moves on the table remaining in contact with the inner wall of the ring. The coefficient of friction between the block and the ring is
m 
. (a) Find the time after which the block will exit the ring at
 B
. (b) With what speed will the block leave the ring at
 B
.
B u A
 
Q. 132. A long helix made of thin wire is held vertical. The radius and pitch of the helix are
 R
 and
r 
 respectively. A bead begins to slide down the helix. (a) Find the normal force applied by the wire on the bead when the speed of the bead is
v
. (b) Eventually, the bead acquires a constant speed of
v
0
. Find the coefficient of friction between the wire and the bead.
R  
r 
 
Q. 133. A wedge of mass
m
 is kept on a smooth table and its inclined surface is also smooth. A small block of mass
m
is projected from the bottom along the incline surface with velocity
u
. Assume that the block remains on the incline and take
q 
= 45°,
g
= 10
m/s
2
.
4
 
 N
EWTON
’
S
 L
 AWS
 
3.25
 (a) Find the acceleration of the wedge and the
 x 
 and
 y
 components of acceleration of the block. (b) Draw the approximate path of the block as observed by an observer on the ground. At what angle does the block hit the table? (c) Calculate the radius of curvature of the path of the block when it is at the highest point.
u m  m 
q 
= 45
y x 
 
Q. 134.
 
A cylinder with radius
 R
 spins about its horizontal axis with angular speed
w 
. There is a small block lying on the inner surface of the cylinder. The coefficient of friction between the block and the cylinder is
m 
. Find the value of
w 
 for which the block does not slip, i.e., stays at rest with respect to the cylinder.
R  
w 
 
Q. 135.
 
A particle of mass
m
 is attached to a vertical rod with two inextensible strings
 AO
 and
 BO
 of equal lengths
l
. Distance between
 A
 and
 B
 is also
l
. The setup is rotated with angular speed
w 
 with rod as the axis.
O m l l l AB 
 (a) Find the values of
w 
 for which the particle remains at point
 B
. (b) Find the range of values of
w 
 for which tension (
T 
1
) in the string
 AO
 is greater than mg but the other string remains slack  (c) Find the value of
w 
 for which tension (
T 
1
) in string
 AO
 is twice the tension (
T 
2
) in string
 BO
 (d) Assume that both strings are taut when the string
 AO
 breaks. What will be nature of path of the particle moment after
 AO
 breaks ?
 
Q. 136. A sphere of mass
m
 and radius
r 
 = 3
m
 is placed inside a container with flat bottom and slant sidewall as shown in the figure. The sphere touches the slant wall at point
 A
 and the floor at point
 B
. It does not touch any other surface. The container, along with the sphere, is rotated about the central vertical axis with angular speed
w 
. The sphere moves along with the container, i.e., it is at rest relative to the container. The normal force applied by the bottom surface and the slant surface on the sphere are
 N 
1
 and
 N 
2
 respectively. There is no friction. (a) Find the value of
w 
 above which
 N 
2
 becomes larger than
 N 
1
 (b) Find the value of
w 
 above which the sphere leaves contact with the floor.
r AB 
 60°2.0
 m 
w 
 
Q. 137. A car is being driven on a tilted ground. The ground makes an angle
q 
 with the horizontal. The driven drives on a circle of radius
 R
. The coefficient of friction between the tires and the ground is
m 
. (a) What is the largest speed for which the car will not slip at point
 A
? Assume that rate of change of speed is zero. (b) What is the largest constant speed with which the car can be driven on the circle without slipping?
AO B 
q 
 
Q. 138. A particle
P
 is attached to two fixed points
O
1
 and
O
2
 in a horizontal line, by means of two
4
 
 3.26
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
light inextensible strings of equal length
l
. It is projected with a velocity just sufficient to make it describe a circle, in a vertical plane, without the strings getting slack and with the angle <
O
2
O
1
P
 = <
O
1
O
2
P
 =
q 
. When the particle is at its lowest point, the string
O
2
P
 breaks and the subsequent path of the particle was found to be a circle of radius
l
 cos
q 
. Find
q 
.
O 
1
 O 
2
P 
q q 
 
Q. 139. The arrangement shown in figure is in equilibrium with all strings vertical. The end
 A
 of the string is tied to a ring which can be slid slowly on the horizontal rod. Pulley
P
1
 is rigidly fixed but
P
2
 can move freely. A mass
m
 is attached to the centre of pulley
P
2
 through a thread. Pulleys and strings are mass less.
AP 
1
P 
2
m M 
 (a) Which block will move up as
 A
 is moved slowly to the right? (b) Will the block of mass
m
 have horizontal displacement? (c) Is it possible, for a particular position of
 A
, that
 M 
 has no acceleration but
m
 does have an acceleration? If this happens when string from
P
2
 to
 A
 makes an angle
q 
 with vertical, find the acceleration of
m
 at the instant. Q. 140. A smooth spherical ball of mass
 M 
 = 2
kg
 is resting on two identical blocks
 A
 and
 B
 as shown in the figure. The blocks are moved apart with same horizontal velocity
V 
 = 1
m/s
 in opposite directions (see figure). (a) Find the normal force applied by each of the blocks on the sphere at the instant separation between the blocks is
a
 =
 2R
;
 R
 = 1.0
m
 being the radius of the ball.
V A B C 
a
 V 
 (b) How much force must be applied on each of the two blocks (when
a
 = 2
 R
) so that they do not have any acceleration. Assume that the he horizontal surface is smooth. Q. 141. In the figure all pulleys (
P
1,
P
2,
P
3 …….) are massless and all the blocks (1,2,3 …..) are identical, each having mass
m
. The system consist of infinite number of pulleys and blocks. Strings are light and inextensible and horizontal surfaces are smooth. Pulley
P
1 is moved to left with a constant acceleration of
a
0
. Find the acceleration of block1. Assume the strings to remain horizontal.
P1
123
P2 P3 a 
0
 Q. 142. A small disc
P
 is placed on an inclined plane forming an angle
q 
 with the horizontal and imparted an initial velocity
v
0
. Find how the velocity of disc depends on the angle
f 
 which its velocity vector makes with the
 x 
 axis (see figure). The coefficient of friction is
m 
= tan
q 
 and initially
f  p 
0
2
=
.
P 
f q 
v x 
5
 
 N
EWTON
’
S
 L
 AWS
 
3.27
ANSWERS
 (a) straight line (b) Parabolic 20
m
3.
43
 H g
 
 N 
 = 12; Tension =
F  N F 
=
12
5.
16
 N 
6.
 (i) (a) True (b) True (ii)
g
5 (iii)
21
ηη
hg
()
−
 7.
t  M m L Mg
=−
2() (a) 
a g
=
4 (b) 
a g
=
5 4
0
 Mmg M m M g
++
 
m m mm m
01212
4
=+
;
 M  m mm m
=+
8
1212
 All masses will fall down with acceleration
g
11.
 (a) More than 9/2
 Mg
 (b) Tension in
S 
2 =
 Mg
 /2, Tension in
S 
1 = 5
 Mg
 (c) Tension in
S 
2 = Mg/6 
q 
45°,
g
 /2 73.1
 N 
 12.5
s
 T = 2
 Mg
;
 N  Mg
=
2 Zero. 
m 
min
 = 1 (a) 6467 N (b) 22400
 N 
 (c) 190400
 N 
 (d)
T 
0
and
F 
 do not change.
T 
 will increase.
19.
 (a) No (b) 1 : 4 22
hg
 
a
 A
 =
a
 B
=
F m
2;
a
c
 = 0
22.
 3
 M 
 /5
q 
 = 62.5°
4.
 R
 A
 
:
 R
 B
:
 R
C
= 3 : 1 : 2 
K 
 = 2 Sdg
26.
 (a) 53
1
cms
–
 (b) 6
 N 
27.
(a) 2, 4 (b) In both cases acceleration of the frame must be ‘
g
’.15
m
 636
t 
–
p 
30.
 
m 
 £
 vgL
02
31.
 4
kg
 4.8
kg
 174
 N 
 1.5
kg
 
m
 9.5
kg
 5
F m
(a)At
C  
(b) At
C  
(c) 2
m/s
2
 and 2
m/s
2
 (i) 15 <
w 
16.67
rad 
 / 
s
 (ii) 500
rad 
 
s
–1
5
 
 3.28
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
 (a) 815
0
gt 
 (b)
gt 
0
o t
8—15
gt 
0
——52
t 
0
 3
t 
0
t 
0
 (a) 80
 N 
 (b) 6409
 N 
 for both(a)
mm
 A B
=
13 (b)
 X 
min
 = 0.75
m
 (c)
¢ = +Ê Ë Áˆ ¯ ˜  ¢ = -Ê Ë Áˆ ¯ ˜ 
- -
V ms V ms
 A B
32323212
11
;
 
V  Mgn
0
4431
= Ê Ë Á ˆ ¯ ˜  -ÈÎ͢˚˙
m 
󰁬
43.
t 
2
 
 N  Mg
1222
12
=+
cossin
q q 
 
a g
=+
312
2
sinsin
q q 
 1.0
s
, 1.0
m
 (b)
T 
P
 = 21.65
 N 
 (c) 3.05
kg
 52221–– 
t  d g
=
12sinsin
θ φ
49.
 52
 MgK 
50.
 2.9
s
51.
 (a) Force between the wall and the middle ball is maximum. It is 4
mg
 (b) Force between upper ball and wall is least. It is 43 
mg
. 
m  q q q 
min
sincostan
=+
22
(a)
K 
 = 2.5
 N 
 / 
cm
 (b) No (a) The block is at height
h
 = 2.5
m
 (b)
V 
 = 5 2
m/s
 (c) 25
m/s
2
 () Zero (a) 2
F K 
 (b)
F K 
 With pulley
P
1
 having zero mass, equilibrium is not possible 
l
3
 (a)
 M 
 (b)
g
3. (a)
q  p 
0
4
=
 (b)
a g
max
cossin
= Ê Ë Á ˆ ¯ ˜  - Ê Ë Á ˆ ¯ ˜ ÈÎ͢˚˙
p p 
88 (a)
t 
 >
t 
1
 (b)
q 
 = Ê Ë Á ˆ ¯ ˜ 
-
tan
1
112
 
a
0
 cos
q 
 
t s
103
 (a)
g
 sin
q 
 (b)
=
122
mg
sin
 q 
 
 N 
 =
mg
 cos
2
 
q 
 (c)
t h M m M m g
= +
( )
+
( )
2
22
sinsin
q q 
 
5
 
 N
EWTON
’
S
 L
 AWS
 
3.29
65.
 (a)
a a
 A B
 
⋅ =
0
 immediately after release (b)
a g
 A
= ←
( )
2 (a)
q 
 (b)
tansincossin
-
 ◊-Ê Ë Á ˆ ¯ ˜ 
12
2
q q q 
 (a) 2
mg
 (b)
t  Lg
=
32
68.
 
gag
sincoscossinsincoscossin
q m q q m q q m q q m q 
-
( )
+
( )
 £ £ +
( )
-
( )
 
m M 
= =
20334167–. (a) (
 M 
 +
m
)
g
 sin
a 
 (b)
 M m gm M 
+
( )
+
sinsin
22
a a 
 
 M m
=
15 
a g
0
48199
=
 44205
g
74.
25
 L
 (a)
d x dt g
222
32
=
 (b)
d x dt g d x dt g
212222
22
= =
; (c)
d x dt g d x dt g
212222
232;
 (a) 4
12012
k k x k k M 
+
( )
 (b) 4
12012
k k x k k 
+
 
a g
=
2;
 N 
 AB
 = 0
78.
 3 647
g
 2
F 
0
 
T 
 = 0.49
 N 
 (a) Zero (b)
F mF m
12
3
=
 (c) To right (a)
q 
13° (b) 37° (a) 137
s
 (b) 0.18 
V  u ugL
max
 = -+ +
 ( )
242
242
m 
 
k 
 = 130
 Nm
–1
;
m 
 = 0.5 
m 
 =+Ê Ë Áˆ ¯ ˜ 
3443
00
a ga g
–
 
r  R
Ê Ë Á ˆ ¯ ˜  = + -+ +
min
11
22
m m m m 
 
F  Mg
=- + +-ÈÎ͢˚˙
1
2
m m  m q q q m q 
cossincossin
 5
m/s
2
 
a
 7
m/s
2
 2
kg
 
m
 30
kg
 (a)
96095
kg
 (b) 48061
kg
 
m q 
min
tan
= D
mm
2
5
 
 3.30
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
 (i) 2.5
kg
; 12.5
 N 
 (ii) 503
 N 
 (iii)
40353
 N a
,
 
m/s
2
 (iv)
56
2
m s
 / 
 
 M R
0
21
=       λ  π
– 
 H  x 
 (a)
m 
 = ++ =
313504.
 (b)
T X A B  C  D  E 
0.17
 Mg 
0.1
 Mg 
0.07
 Mg 
 (c)
T 
max
 = 0.17
 Mg
 (i) (a)
T 
 BC 
 = 10
 N 
;
T 
 AB
 = 12
 N 
;
T 
 AD
 = 7
 N 
 (b)
C 
 = 0;
 B
 = 4
 N 
;
 A
 = 5
 N 
;
 D
 =
5N 
 (ii) (a)
T 
100
11
=+Ê Ë Áˆ ¯ ˜ 
–
 m m 
 T
0 
(b)
m M 
20020
11
=+Ê Ë Áˆ ¯ ˜ 
–
 m m 
 
w 
 =
 g R
 
w q q 
= -
( )
g R
cossin2
 334
mgm M 
+
102.
 
 x l
=
0
31
h
–
103.
 2.45 (a)
q m m 
010202
1
=-+Ê Ë ÁÁÁÁˆ ¯ ˜ ˜ ˜ ˜ 
-
tan
V  RgV  Rg
 (b)
V rg
10
=
tan
q 
 (c) Obtuse (a) 22° (b) (i) 2315
 N 
, 1389
 N 
 (a) 
t  R g f V  f 
022220
14
=−
( )
  −µ
 (b)
mV ft  R f 
0422
+
( )
+
 (a) 93.3 N (b) 107 
m/s
2
 (c) 4
m/s
 
F m
5
109.
 (a) contact force = 0, acceleration of 4
kg
 block is 0.7
m/s
2
 and that of other block is zero
 (b) contact force = 1.4
 N,
 acceleration of both = 0 (c ) Contact force = 5.74
 N,
acceleration of both = 1.27
m/s
2
 
q 
 = Ê Ë Áˆ ¯ ˜ 
-
cos
1
341
111.
 30.1
s
 (a)
g R
231331–
( )
+
( )
 (b)
g R
 23 (a)
F  R g
022
12
= -
( )
sp q q q 
cossin.cos
5
 
 N
EWTON
’
S
 L
 AWS
 
3.31
 (b)
F 
0
 = 0 (c)
 R
 = r (1 + 2 7)
116.
 54
0
l
 (a)
a g M m
11
14
=      
– (b)
m kg
4
1811
>
 
F 
0
 = 2
m 
m
2
g
 
m mm m
2121
2
++    
; increase
119.
 Zero (a)
q 
= 45° from vertical diameter. (b)
T Mg
max
 = -Ê Ë Áˆ ¯ ˜ 
221
p 
121.
 Zero 21° <
q 
< 33° 
O2 12.67 23.3316-1634
t 
 (s)
f 
A
 (N)
 
20434
t 
 (s)(N)O2 12.67
f 
B 
124.
 
q a 
= +Ê Ë Á ˆ ¯ ˜ ÈÎ͢˚˙
-
21
1
tancot
mgF 
 (a)
T 
0
= 2
 R
g (b) At the lowest point sin
−
     
1
13
127.
 
F  mg
min
 = −
( )
2231
128.
 (a)
F Mg
max
 =
34
m 
 (b) Between
 A
 and
 B
 (c) 2, No
129.
 (a) 2
ms
–1
 (b) 2.1
kg
 (a)
T 
 =
 N 
 cos
q 
 (b)
mg M m
tancot
q q 
+
 (a)
t  Rue
= -ÈΠ˘˚
m 
pm 
1
 (b)
V  ue
=
πµ
 (a)
mg v Rg
coscos
q  q 
1
22
+ Ê Ë Áˆ ¯ ˜ 
 (b)
tancos
q q 
1
22
+ Ê Ë Áˆ ¯ ˜ 
v Rg
 where tan
q  r p 
=
2
 R
133.
 (a)
a gi
wedge
=
3ˆ 
a g
 x block 
 =
–3 
a g
 y block 
 =
23 (b) The block hits the table normally. (c) 316
2
ug
 
w  m m 
≥ +
g R
1
2
5
 
 3.32
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
135.
 (a)
w 
 >
 gl
 (b)
glgl
< £
w 
2 (c) 6
gl
 (d) parabolic (a)
g
3 (b) 3
g
. (a) [
g
2
 
 R
2
 (
m 
2
 cos
2
 
q 
– sin
2
 
q 
)]
1/4
 (b)
gR
 m q q 
cossin
-
( )
SOLUTIONS
1.
 (a) Initial velocity is parallel to
F 
 or anti parallel to
F 
. Hence particle moves in a straight line and speed may increase or decrease. (b) Path is parabolic with speed increasing. In case (a) the particle may retrace its path.
2.
 Just before striking the ceiling, retardation is 2
g
. If air resistance force is
 R
at this instant, then
v v R R 
a
mg mg 
 
ma
 =
mg
 +
 R
 
m
 (2
g
) =
mg
 +
 R
 
 R
 =
mg
 After impact, the air resistance force will be upward but its magnitude will remain mg. This is because speed has not changed.After impact net force on the ball = 0Ball will fall down with constant speed
 H 
 = (10
m/s
) (2
s
) = 20
m
. 
q 
 = Ê Ë Áˆ ¯ ˜ 
-
tan
1
15
139.
 (a) Block with mass
 M 
 will move up. (b) yes (c)
g
 (1 – cos
q 
) (a) (10 2 – 8)
 N  
(b) (5 – 42)
 N 
 32
0
a
 
v
0
1
+
cos
f 
5
 
L 
EVEL 
 1
 
Q. 1. (i) The cause of increases in kinetic energy when a man starts running without his feet slipping on ground is asked to two students. Their answers are– Harshit: Cause of increase in kinetic energy is work done by friction force. Without friction the man cannot run. Akanksha: Cause of increase in kinetic energy is work done by internal (muscle) forces of the body. Who is right? (ii) An inextensible rope is hanging from a tree. A monkey, having mass
m
, climbs to a height
h
 grabbing the rope tightly. The monkey starts from rest and ends up hanging motionlessly on the rope at height
h
. (a) How much work is done by gravity on the monkey? (b) How much work is done by the rope on the monkey? (c) Using work – energy theorem, explain the increase in mechanical energy of the monkey.
 
Q. 2. A man of mass
 M 
 jumps from rest, straight up, from a flat concrete surface. Centre of mass of the man rises a distance h at the highest point of the motion. Find the work done by the normal contact force (between the man’s feet and the concrete floor) on the man.Q. 3.A block of mass
m
 = 10
kg
 is released from the top of the smooth inclined surface of a wedge which is moving horizontally toward right at a constant velocity of
u
 = 10
m/s
. Inclination of the wedge is
q 
= 37°. Calculate the work done by the force applied by the wedge on the block in two seconds in a reference frame attached to -
WORK - POWER - ENERGY 
m u 
 (a) the ground (b) the wedge. [Take
g
 = 10 ms
–2
]
 
Q. 4. In an industrial gun, when the trigger is pulled a gas under pressure is released into the barrel behind a ball of mass
m
. The ball slides smoothly inside the barrel and the force exerted by the gas on the ball varies as Where
 L
 is length of the end of the barrel from the initial position of the ball and
 x 
 is instantaneous displacement of the ball from its initial position. Neglect any other force on the ball apart from that applied by the gas. Calculate the speed (
V 
) of the ball with which it comes out of the gun.
L
xGas Cy linder 
Q. 5. A particle of mass 3
kg
 takes 2 second to move from point
 A
 to
 B
 under the action of gravity and another constant force 
( )
= - +
, where the unit vector ˆ
k  
is in the direction of upward vertical. The position vector of point
 B
 is
( )
= - -
 and velocity of the particle when it reaches
 B
is 
( )
= + -
. (a) Find the velocity,
 of the particle when it
04
5
 
 4.2
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
was at
 A
. (b) Find position vector,
of point
 A
. (c) Find work done by the force
 as the particle moves from
 A
to
 B
. (d) Find change in gravitational potential energy of the particle as it moves from
 A
 to
 B
.
 
Q. 6. A particle can move along a straight line. It is at rest when a force (
F 
) starts acting on it directed along the line. Work done by the force on the particle changes with time(t) according to the graph shown in the fig. Can you say that the force acting on the particle remains constant with time?
W  O  t 
 
Q. 7. A particle is moving on a straight line and all the forces acting on it produce a constant power
P
 calculate the distance travelled by the particle in the interval its speed increase from
V 
 to 2
V 
.
 
Q. 8. Work done and power spent by the motor of an escalator are
W 
 and
P
 respectively when it carries a standing passenger from ground floor to the first floor. Will the work and power expended by the motor change if the passenger on moving escalator walks up the staircase at a constant speed?
 
Q. 9. (i) A block is connected to an ideal spring on a horizontal frictionless surface. The block is pulled a short distance and released. Plot the variation of kinetic energy of the block vs the spring potential energy. (ii) A ball of mass 200
g
 is projected from the top of a building 20
m
 high. The projection speed is 10 m/s at an angle
q 
 -
 Ê ˆ = Á ˜ Ë ¯ 
 from the horizontal. Sketch a graph of kinetic energy of the ball against height measured from the ground. Indicate the values of kinetic energy at the top and bottom of the building and at the highest point of the trajectory, specifying the heights on the graph. Neglect air resistance and take
g
 = 10
m/s
2
 
Q. 10. A car of mass
m
 = 1600
kg
, while moving on any road, experiences resistance to its motion given by (
m
 +
nV 
2
) newton; where
m
 and
n
 are positive constants. On a horizontal road the car moved at a constant speed of 40
m
 / 
s
 when the engine developed a power of 53
KW 
. When the engine developed an output of 2
KW 
 the car was able to travel on a horizontal road at a constant speed of 10
m
 / 
s
. (a) Find the power that the engine must deliver for the car to travel at a constant speed of 40
m/s
 on a horizontal road. (b) The car is able to climb a hill at a constant speed of 40
m
 / 
s
 with its engine working at a constant rate of 69
KW 
. Calculate the inclination of the hill (in degree)
 
Q. 11.
 
A particle moves along the loop
 A
–
 B
–
C 
–
 D
–
 A
 while a conservative force acts on it. Work done by the force along the various sections of the path are –
W 
 A
 
"
 
 B
 = – 50
 J 
 ;
W 
 B
 
"
 
C 
 = 25
 J 
; W
C 
"
 D
=
60
 J 
. Assume that potential energy of the particle is zero at A. Write the potential energy of particle when it is at B and D.
ADC
 
Q. 12. A moving particle of mass m is acted upon by five forces

 and

. Forces

and
F 
3
 are conservative and their potential energy functions are
U 
 and
W 
 respectively. Speed of the particle changes from
V 
a
 to
V 
b
 when it moves from position
a
 to
b
. Which of the following statement is/are true – (a) Sum of work done by

 and

 = 
U 
b
 –
U 
a
 +
W 
b
 –
W 
a
 (b) Sum of work done by

 and
 = (c) Sum of work done by all five forces = (d) Sum of work done by

 and

 = (
U 
b
 +
W 
b
) – (
U 
a
 +
W 
a
). Q. 13.
1O-112.53.55.5X (in m)F (in N)
5
 
 W
ORK
 - P
OWER
 - E
NERGY
 
4.3
 
The given graph represents the total force in
 x 
 direction being applied on a particle of mass
m
 = 2
kg
 that is constrained to move along
 x 
 axis. What is the minimum possible speed of the particle when it was at
 x
= 0?
 
Q. 14.
 
A vertical spring supports a beaker containing some water in it. Water slowly evaporates and the compression in the spring decreases. Where does the elastic potential energy stored in the spring go?Q. 15.A pan of negligible mass is supported by an ideal spring which is vertical. Length of the spring is
 L
o
. A mass
 M 
 of sand is lying nearby on the floor. A boy lifts a small quantity of sand and gently puts it into the pan. This way he slowly transfers the entire sand into the pan. The spring compresses by . Assume that height of the sand heap on the floor as well as in the pan is negligible. Calculate the work done by the boy against gravity in transferring the entire sand into the pan.
L
0
M 
 
Q. 16. A snake of mass
 M 
 and length
 L
 is lying on an incline of inclination 30°. It craws up slowly and overhangs half its length vertically. Assume that the mass is distributed uniformly along the length of the snake and its hanging part as well as the part on the incline both remain straight.
L  
30°
L
2
 (a) Find the work done by the snake against gravity (
W 
g
) (b) Will the answer to part (a) be different if the snake were of half the length but of same mass. Q. 17.
 
A uniform rope of linear mass density
l 
 (
kg
 / 
m
) passes over a smooth pulley of negligible dimension. At one end B of the rope there is a small particle having mass one fifty of the rope. Initially the system is held at rest with length
 L
 of the rope on one side and length on the other side of the pulley (see fig). The external agent begins to pull the end A downward. Find the minimum work that the agent must perform so that the small particle will definitely reach the pulley. 
AForceby external agent
L4LB
 
Q. 18. A particle of mass
m
 = 100
g
 is projected vertically up with a kinetic energy of 20
 J 
 form a position where its gravitational potential energy is – 50
 J 
. Find the maximum height to which the particle will rise above its point of projection. [
g
 = 10
m/s
2
] Q. 19. A physics student writes the elastic potential energy stored in a spring as
= +
, where
 L
 is the natural length of the spring,
 x 
 is extension or compression in it and
K 
 is its force constant. A block of mass
 M 
 travelling with speed
V 
 hits the spring and compresses it.
LVM
 Find the maximum compression caused.
 
Q. 20. A block of mass
m
= 4
kg
 is kept on an incline connected to a spring (see fig). The angle of the incline is
q 
 = 30° and the spring constant is
5
 
 4.4
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
K 
 = 80
 N/m
. There is a very small friction between the block and the incline. The block is released with spring in natural length. Find the work done by the friction on the block till the block finally comes to rest.[
g
 = 10
m/s
2
]
= 30°
 m k
 
Q. 21. A body is projected directly up a plane which is inclined at an angle
q 
 to the horizontal. It was found that when it returns to the starting point its speed is half its initial speed. (a) Was dissipation of mechanical energy of the body, due to friction, higher during ascent or descent? (b) Calculate the coefficient of friction (
m 
) between the body and the incline.
 
Q. 22. A tanker filled with water starts at rest and then rolls, without any energy loss to friction, down a valley. Initial height of the tanker is
h
1
. The tanker, after coming down, climbs on the other side of the valley up to a height
h
2
. Throughout the journey, water leaks from the bottom of the tanker. How does
h
2
 compare with
h
1
?
h 
1
 
Q. 23. A stone with weight
W 
 is thrown vertically upward into air with initial speed
u
. Due to air friction a constant force
 R
 acts on the stone, throughout its flight. Find – (a) the maximum height reached and (b) speed of stone on reaching the ground.
 
Q. 24. A mass
m
= 0.1
 kg
 is attached to the end
 B
 of an elastic string
 AB
 with stiffness
k = 16 N/m
 and natural length
l
0
=
0.25
 m
. The end
 A
 of the string is fixed. The mass is pulled down so that
 AB
 is
2l
0
=
0.5
 m
 and then released. (a) Find the velocity of the mass when the string gets slack for the first time. (b) At what distance from
 A
 the mass will come to rest for the first time after being released.
AB 
 Q. 25. Two blocks 1 and 2 start from same point
 A
 on a smooth slide at the same time. The track from
 A
 to
 B
 to
C 
 is common for the two blocks. At
C 
 the track divides into two parts. Block 1 takes the route
C–D–E 
 and gets airborne after
 E 
. Block 2 moves along
CFGH 
. Point
 E 
 is vertically above
G
 and the stretch
GH 
 is
 
horizontal. Block 1 lands at point
 H 
. (a) Where is the other block at the time block 1 lands at
 H 
? Has it already crossed
 H 
 or yet to reach there? (b) Which block will reach at
 H 
 with higher speed ?
B A
 2
C F  H 
12
E D G 
1
 Q. 26 In the arrangement shown in the figure, block
 B
 of mass
 M
rests on a weighing scale. Ball
 A
 is released from a position where spring is in its natural length and the scale shows the correct weight of block
 B
. Find the mass of ball
 A
 so that the minimum reading shown by the scale subsequently is half the true weight of
 B
.
k AB  M 
 Q. 27 In an aircraft carrier warship the runway is a 20
m
 long strip inclined at
q 
=
20° to the horizontal. The launcher is effectively a large spring that pushes an aircraft of mass
m =
2000
 
5
 
 W
ORK
 - P
OWER
 - E
NERGY
 
4.5
kg
 for first 5
m
 of the 20
m
 long runway. The jet engine of the plane produces a constant thrust of 6 × 10
4
 
 N 
 for the entire length of the runway. The plane needs to have a speed of 180
kph
 at the end of the runway. Neglect air resistance and calculate the spring constant of the launcher. [sin
 
20° = 0.3 and
g
= 10
m/s
2
] 
  2  0 m
q 
 
Q. 28 A block of mass
 M 
 is placed on a horizontal surface having coefficient of friction
m 
. A constant pulling force
=
 is applied on the block to displace it horizontally through a distance
d 
. Find the maximum possible kinetic energy acquired by the block.
 
Q. 29 A small block is made to slide, starting from rest, along two equally rough circular surfaces from
 A
 to
 B
 through path 1 and 2. The two paths have equal radii. The speed of the block at the end of the slide was found to be
V 
1
 and
V 
2
 for path 1 and 2 respectively. Which one is larger
V 
1
 or
V 
2
?
12AB
 Q. 30 A particle can move along x axis under influence of a conservative force. The potential energy of the particle is given by
U 
 = 5
 x 
2
 –
20
 x +
2 joule where
 x 
 is co-ordinate of the particle expressed in meter. The particle is released at
 x
= –3
m
 (a) Find the maximum kinetic energy of the particle during subsequent motion. (b) Find the maximum
 x 
 co-ordinate of the particle. Q. 31 A particle is constrained to move along
 x 
 axis under the action of a conservative force. The potential energy of the particle varies with position
 x 
 as shown in the figure. When the particle is at
 x
=
 x 
0
, it is given a kinetic energy (
k 
) such that 0 <
k
< 4
U 
0
 (a) Does the particle ever reach the origin? (b) Qualitatively describe the motion of the particle.
U x 
( )6
U 
0
5
U 
0
U 
0
x 
0
x O 
 
Q. 32 A pillar having square cross section of side length
 L
 is fixed on a smooth floor. A particle of mass
m
 is connected to a corner
 A
 of the pillar using an inextensible string of length 3.5
 L
. With the string  just taut along the line
 BA
, the particle is given a velocity
v
 perpendicular to the string. The particle slides on the smooth floor and the string wraps around the pillar.
m 
3.5
 LALLB 
 (a) Find the time in which the particle will hit the pillar. (b) Find the tension in the string just before the particle hits the pillar. Neglect any energy loss of the particle. Q. 33
 
(i) A simple pendulum consist of a small bob of mass
m
 tied to a string of length
 L
. Show that the total energy of oscillation of the pendulum is
 when it is oscillating with a small angular amplitude
q 
0
. Assume the gravitational potential energy to be zero of the lowest position of the bob. (ii) Three identical pendulums
 A, B
 and
C 
 are suspended from the ceiling of a room. They are swinging in semicircular arcs in vertical planes. The string of pendulum
 A
 snaps when
5
 
 4.6
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
it is vertical and it was found that the bob fell on the floor with speed
V 
1
. The string of
 B
 breaks when it makes an angle of 30° to the vertical and the bob hits the floor with speed
V 
2
. The string of pendulum
C 
 was cut when it was horizontal and the bob falls to the floor hitting it with a speed
V 
3
. Which is greatest and which is smallest among
V 
1
,
V 
2
 and
V 
3
? Q. 34
 AB
 is a mass less rigid rod of length 2
l
. It is free to rotate in vertical plane about a horizontal axis passing through its end
 A
. Equal point masses (
m
 each) are stuck at the centre
C 
 and end
 B
of the rod. The rod is released from horizontal position. Write the tension in the rod when it becomes vertical.
A l  l  B C 
 Q. 35 A rigid mass less rod of length
 L
 is rotating in a vertical plane about a horizontal axis passing through one of its ends. At the other end of the rod there is a mass less metal plate welded to the rod. This plate supports a heavy small bead that can slide on the rod without friction. Just above the bead there is another identical metal plate welded to the rod. The bead remains confined between the plates. The gap between the plates is negligible compared to
 L
. The angular speed of the rod when the bead is at lowest position of the circle is
w 
 =
. How many times a clink of  the bead hitting a metal plate is heard during one full rotation of the rod ? 
Bead
 Q. 36 A child of mass
m
 is sitting on a swing suspended by a rope of length
 L
. The swing and the rope have negligible mass and the dimension of child can be neglected. Mother of the child pulls the swing till the rope makes an angle of
q 
0
 = 1 radian with the vertical. Now the mother pushes the swing along the arc of the circle with a force and releases it when the string gets vertical. How high will the swing go? [Take cos(1 radian) ~ 0.5]
 
Q. 37. A particle of mass
m
 is suspended by a string of length
l
 from a fixed rigid support. Particle is imparted a horizontal velocity . Find the angle made by the string with the vertical when the acceleration of the particle is inclined to the string by 45°? Q. 38 A particle of mass m is moving in a circular path of constant radius r such that its centripetal acceleration
a
c
 is varying with time
t 
 as
a
c
 =
k 
2
rt 
2
, where
k 
 is a constant. Calculate the power delivered to the particle by the force acting on it. Q. 39 A ball is hanging vertically by a light inextensible string of length
 L
 from fixed point
O
. The ball of mass
m
 is given a speed u at the lowest position such that it completes a vertical circle with centre at
O
 as shown. Let
 AB
 be a diameter of circular path of ball making an angle
q 
 with vertical as shown. (
g
 is acceleration due to gravity)
AB u O m 
q 
 (a) Let
T 
 A
 and
T 
 B
 be tension in string when ball is at
 A
 and
 B
 respectively, then find
T 
 A
 –
T 
 B
. (b) Let
 and
 be acceleration of ball when it is at
 A
 and
 B
 respectively, then find the value of
. Q. 40 A ball suspended by a thread swings in a vertical plane so that the magnitude of its total acceleration in the extreme position and lowest position are equal. Find the angle
q 
 that the thread makes with the vertical in the extreme position. Q. 41
 
A particle of mass
m
 oscillates inside the smooth surface of a fixed pipe of radius
 R
. The axis of the pipe is horizontal and the particle moves from
 B
 
5
 
 W
ORK
 - P
OWER
 - E
NERGY
 
4.7
to
 A
 and back. At an instant the kinetic energy of the particle is
K 
 (say at position of the particle shown in the figure). What is the force applied by particle on the pipe at this instant?
A B 
L 
EVEL 
 2
 Q. 42.
h R O 
 (i) There is a vertical loop of radius
 R
. A small block of mass
m
 is slowly pushed along the loop from bottom to a point at height
h
. Find the work done by the external agent if the coefficient of friction is
m 
. Assume that the external agent pushes tangentially along the path. (ii) A block of mass
m
 slides down a smooth slope of height
h
, starting from rest. The lower part of the track is horizontal. In the beginning the block has potential energy
U = mgh
 which gets converted into kinetic energy at the bottom. The velocity at bottom is . Now assume that an observer moving horizontally with velocity towards right observes the sliding block. She finds that initial energy of the block is
= +
 and the final energy of the block when it reaches the bottom of the track is zero. Where did the energy disappear?
h m 
 Q. 43.
 
A completely filled cylindrical tank of height
 H 
 contains water of mass
 M 
. At a height
h
 above the top of the tank there is another wide container. The entire water from the tank is to be transferred into the container in time
t 
0
 such that level of water in tank decreases at a uniform rate. How will the power of the external agent vary with time?
h 
H 
 Q. 44
.
 A uniform chain of mass
m
0
 
and length
l
 rests on a rough incline with its part hanging vertically as shown in the fig. The chain starts sliding up the incline (and hanging part moving down) provided the hanging part equals
h
 times the chain length (
h
 < 1). What is the work performed by the friction force by the time chain slides completely off the incline. Neglect the dimension of pulley and assume it to be smooth.
q 
 Q. 45
.
 A large flat board is lying on a smooth ground. A disc of mass
m
 = 2
kg
 is kept on the board. The coefficient of friction between the disc and the board is
m 
 = 0.2 . The disc and the board are moved with velocity
 and
 respectively [in reference frame of the ground]. Calculate the power of the external force applied on the disc and the force applied on the board. At what rate heat is being dissipated due to friction between the board and the disc? [
g
 = 10 ms
–2
]
Y u V 
X 
 Q. 46. A car can pull a trailer of twice its mass up a certain slope at a maximum speed
V 
. Without
5
 
 4.8
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
the trailer the maximum speed of the car, up the same slope is 2
V 
. The resistance to the motion is proportional to mass and square of speed. If the car (without trailer) starts to move down the same slope, with its engine shut off, prove that eventually it will acquire a constant speed. Find this speed. Q. 47 Force acting on a particle in a two dimensional
 XY 
 space is given as
. Show that the force is conservative. Q. 48. In a two dimensional space the potential energy function for a conservative force acting on a particle of mass
m
 = 0.1
kg
 is given by
U 
 = 2 (
 x 
 +
 y
) joule (
 x 
 and
 y
 are in m). The particle is being moved on a circular path at a constant speed of
V 
 = 1
ms
 
–1
. The equation of the circular path is
 x 
2
 +
 y
2
 = 4
2
. (a) Find the net external force (other than the conservative force) that must be acting on the particle when the particle is at (0, 4). (b) Calculate the work done by the external force in moving the particle from (4, 0) to (0, 4). Q. 49. A particle of mass
m
 moves in
 xy
 plane such that its position vector, as a function of time, is given by
; where
b
and
k 
 are positive constants. (a) Find the time
t 
0
 in the interval when the resultant force acting on the particle has zero power. (b) Find the work done by the resultant force acting on the particle in the interval Q.50. A block of mass 2 kg is connect to an ideal spring and the system is placed on a smooth horizontal surface. The spring is pulled to move the block and at an instant the speed of end A of the spring and speed of the block were measured to be 6
m/s
 and 3
m/s
respectively. At this moment the potential energy stored in the spring is increasing at a rate of 15
 J/s
. Find the acceleration of the block at this instant.
A2kg
 Q. 51.
 
A body of mass
m
is slowly hauled up a rough hill as shown in fig by a force
F 
 which acts tangential to the trajectory at each point. Find the work performed by the force, if the height of hill is
h
, the length of its base
l
 and coefficient of friction between the body and hill surface is
m 
. What is the work done if body is moved along some alternative path shown by the dotted line, friction coefficient being same.
 m F
l
h 
 Q. 52. In previous problem what is the work done by
if the body started at rest at the base and has a velocity
v
on reaching the top?
 
Q. 53. A block of mass
 M 
 is placed on a horizontal smooth table. It is attached to an ideal spring of force constant
k 
 as shown. The free end of the spring is pulled at a constant speed
u
. Find the maximum extension (
 x 
o
) in the spring during the subsequent motion.
u M k 
 
Q. 54.
 
A spring block system is placed on a rough horizontal floor. Force constant of the spring is
k 
. The block is pulled to right to give the spring an elongation equal to
 x 
0
 and then it is released. The block moves to left and stops at the position where the spring is relaxed. Calculate the maximum kinetic energy of the block during its motion.
 
Q. 55.
 
In the fig shown, a block of mass
 M
is attached to the spring and another block of mass 2
 M 
 has been placed over it. The system is in equilibrium. The block are pushed down so that the spring compresses further by . System is released.
2
M M k 
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
2
3
3
4
3
3
 
 M
OMENTUM
 
 AND
 C
ENTER
 
OF
 M
 ASS
 
5.15
ground with the bottom of the lower ball at height
h
 above the ground. The lower ball has a radius
 R
 and the upper ball has negligible dimension.
m M 
h 
 (a) Up to what height the ball of mass
m
 will bounce above the ground ? (b) Does the result obtained above violates the low of conservation of mechanical energy?
 
Q. 83. Three identical particles are placed on a horizontal smooth table, connected with strings as shown. The particle
 B
 is imparted a velocity
V 
0
 = 9
m/s
 in horizontal direction perpendicular to the line
 ABC 
. Find speed of particle A when it is about to collide with C.
V 
0
B l A l C 
 
Q. 84. A light inextensible string, passing over a pulley, supports two particles 1 and 2 at its ends. An insect of mass m is sitting on particle 2 and the system is in equilibrium. The sum of masses of particles and the insect is
 M 
. Now the insect crawls a distance
 x 
 up relative to the string. Find the displacement of centre of mass of the system of two particles and the insect. In which direction does the centre of mass move and why?
 
Q. 85. Two particles (A and B) of masses m and 2m are  joined by a light rigid rod of length L. The system lies on a smooth horizontal table. The particle (A) of mass m is given a sharp impulse so that it acquires a velocity u perpendicular to the rod. Calculate maximum speed of particle B during subsequent motion. By what angle
q 
 will the rod rotate by the time the speed of particle B become maximum for the first time?
12
LB Au 
 
Q. 86. Two blocks
 A
 and
 B
, each of mass m, are connected by a spring of force constant
K 
. Initially, the spring is in its natural length. A horizontal constant force
F 
 starts acting on block
 A
 at time
t 
=0 and at time
t 
 , the extension in the spring is seen to be
󰁬
. What is the displacement of the block
 A
 in time
t 
?
AF B 
 
Q. 87. Two blocks of mass
m
1
 and
m
2
 are connected to the ends of a spring. The spring is held compressed and the system is placed on a smooth horizontal table. The block of mass
m
1
 = 2
kg
 is kept at
 x 
 = 1 cm mark and the other block is at
 x 
 = 2 cm mark. The system is released from this position. It was observed that at the instant
m
1
 was at
 x 
 = 5 cm mark its velocity was zero and at that moment
m
2
 was located at
 x 
 = –4 cm. Find mass
m
2
 andunstretched length (
l
0
)of the spring.
-5 -4 -3 -2 -1 0 1 2 3 4
x 
5
m 
2
 m 
1
k 
 
Q. 88. Two particles having masses
m
1
 and
m
2
 are moving with velocities
V 
1
 and
V 
2
 respectively.
V 
0
 is velocity of centre of mass of the system. (a) Prove that the kinetic energy of the system in a reference frame attached to the centre of mass of the system is
KE 
cm rel
V 
=
12
2
µ
. Where
µ =+
m mm m
1212
 and
V 
rel
 is the relative speed of the two particles. (b) Prove that the kinetic energy of the system in ground frame is given by
KE KE m m V 
cm
= + +
( )
12
1202
 (c) If the two particles collide head on find the minimum kinetic energy that the system has during collision.
 
Q. 89.
 
Two blocks
 A
 and
 B
 of mass m and 2m respectively are connected by a light spring of force constant
k 
. They are placed on a smooth horizontal surface. Spring is stretched by a length
 x 
 and then released. Find the relative velocity of the blocks when the spring comes to its natural
5
 
 5.16
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
length. 
B A
 
Q. 90. Two ring of mass
m
 and 2
m
 are connected with a light spring and can slide over two frictionless parallel horizontal rails as shown in figure. Ring of mass m is given velocity ‘
v
0
’ in horizontal direction as shown. Calculate the maximum stretch in spring during subsequent motion.
m  v 
0
k 
2
m 
 
Q. 91. A disc of mass
 M 
 and radius
 R
 is kept flat on a smooth horizontal table. An insect of mass m alights on the periphery of the disc and begins to crawl along the edge. (a) Describe the path of the centre of the disc. (b) For what value of
m M 
 the centre of the disc and the insect will follow the same path? 
O R 
 
Q. 92. A metal wire having mass
 M 
 is bent in the shape of a semicircle of radius
 R
 and is sliding inside a smooth circular grove of radius
 R
 present in a horizontal table. The wire just fits into the groove and is moving at a constant speed
V 
. Find the magnitude of net force acting on the wire.
R V 
 
Q. 93. A triangular wedge (A) has inclined surface making an angle
q 
 = 37° to the horizontal. A motor (
 M 
) is fixed at the top of the wedge. Mass of the wedge plus motor system is 3
m
. A small block (B) of mass
m
 = 1kg is placed at the bottom of the incline and is connected to the motor using a light string. The motor is switched on and it slowly hauls block B through a distance
 L
 = 2.0 meter along the incline. Calculate the work done by the string tension force on the wedge plus motor system. All surfaces are frictionless.
q 
A B
L 
M
 
Q. 94. An ice cream cone of mass
 M 
 has base radius
 R
 and height
h
. Assume its wall to be thin and uniform. When ice cream is filled inside it (so as to occupy the complete conical space) its mass becomes 5
 M 
. Find the distance of the centre of mass of the ice cream filled cone from its vertex.
 
Q. 95. A flexible rope is in the shape of a semicircle ACB with its centre at O. Ends
 A
 and
 B
 are fixed. Radius of the semicircle is
 R
. The midpoint C is pulled so that the rope acquires
V 
 shape as shown in the figure.
O R C B AA O B C 
 (a) Make a guess whether the centre of mass of the rope moves closer to O or moves away from it when it is pulled? (b) Calculate the shift in position of the centre of mass of the rope. Q. 96. Three small balls of equal mass (
m
) are suspended from a thread and two springs of same force constant (
K 
) such that the distances between the first and the second ball and the second third ball are the same. Thus the centre of mass of the whole system coincides with the second ball. The thread supporting the upper ball is cut and system starts a free fall. Find the distance of the centre of mass of
m m k m m k 
5
 
 M
OMENTUM
 
 AND
 C
ENTER
 
OF
 M
 ASS
 
5.17
the system from the second ball when both the springs acquire their natural length in the falling system.
 
Q. 97.
 
(a) A uniform chain is lying in form of on arc of a circle of radius R. The arc subtends an angle of 2 at the centre of the circle. Find the distance of the centre of mass of the chain from the centre of the circle.
O
R 
a a 
   O
   O
   O
  O
  O
 O
 O
 O
 O
O
O
O
O
O 
O 
O 
O 
O  
O  
O  
O   
O   
O   
 (b) A uniform chain of length
π
 R
2 is lying symmetrically on the top of a fixed smooth half cylinder (see figure) of radius
 R
. The chain is pulled slightly from one side and released. It begins to slide. Find the speed of the chain when its one end just touches the floor. What is speed of centre of mass of the chain at this instant? 
RO
   O
   O
   O
  O
  O
 O
 O
 O
O
O
O
O 
O 
O 
O  
O  
O  
O   
O   
O   
 (c) In part (b) assume that the half cylinder is not fixed and can slide on the smooth floor. Find the displacement of the cylinder by the time one end of the chain touches the floor. Mass of cylinder is equal to that of the chain. For part (b) and (c) assume that the chain remains in contact with the cylinder all the while.
 
Q. 98. A small body of mass m is at rest inside a narrow groove carved in a disc. Groove is a circle of radius R concentric to the disc. Mass of the disc is also m . The disc lies on a smooth horizontal floor. The small body is given a sharp impulse so that it acquires a tangential velocity
V 
o
 at time
t 
 = 0.
grooveDiscV
o
R
 (a) The velocity of the centre of the disc becomes zero for the first time at time
t 
0
. Find
t 
0
. (b) Find speed of the small body at time
t 
0
3
.
 
Q. 99. Laila and Majnu are on a boat for a picnic. The boat is initially at rest. Laila has a big watermelon which she throws towards Majnu. The man catches the melon and eats half of it. He throws back the remaining half to Laila. She eats the half of the melon that she receives & throws the remaining part to Majnu. Majnu again eats half of what he receives and returns the remaining part back to Laila. This continues till the melon lasts. The two are sitting at the two ends of the boat which has a length L. Combined mass of the boat and the two lovers is
 M 
0
 and the mass of the water melon is M. Assume that the boat can move horizontally on water without any resistive force. Find the displacement of the boat when the watermelon gets finished. 
L
 Q. 100. A hot air balloon (mass M) has a passenger (mass m) and is stationary in the mid air. The passenger climbs out and slides down a rope with constant velocity u relative to the balloon. (a) Show that when the passenger is sliding down, there is no change in mechanical energy (kinetic + gravitational potential energy) of the system (Balloon + passenger). Calculate the speed of balloon. (b) Calculate the power of the buoyancy force on the system when the man is sliding. For easy calculation, assume that volume of man is negligible compared to the balloon. (c) If buoyancy force is doing positive work, where is this work done lost? You have proved that sum of kinetic and potential energy of the system remains constant.
Mm
5
 
 5.18
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
 Q.101. A wooden wedge of mass 10
m
 has a smooth groove on its inclined surface. The groove is in shape of quarter of a circle of radius
 R
 = 0.55 m. The inclined face makes an angle 
q 
 = Ê Ë Áˆ ¯ ˜ 
-
cos
1
115
with the horizontal. A block  ‘
 A
’ of mass
m
 is placed at the top of the groove and given a gentle push so as to slide along the groove. There is no friction between the wedge and the horizontal ground on which it has been placed. Neglect width of the groove.
ARR
q 
 (a) Find the magnitude of displacement of the wedge by the instant the block A reaches the bottom of the groove. (b) Find the velocity of the wedge at the instant the block A reaches the bottom of the groove. Q. 102. A uniform bar
 AB
 of length 6
a
 has been placed on a horizontal smooth table of width 5
a
 as shown in the figure. Length 2
a
 of the bar is overhanging. Mass of the bar is 4
m
. An insect of mass
m
 is sitting at the end
 A
 of the bar. The insect walks along the length of the bar to reach its other end
 B
.
m A
4
m 
5
a
6
a
2
a
B 
 (a) Will the bar topple when the insect reaches end
 B
 of the bar? (b) After the insect reaches at
 B
, another insect of mass
 M
lands on the end
 A
 of the bar. Find the largest value of
 M 
 which will not topple the bar.
 
Q. 103. A disc of mass
 M 
 and radius
 R
lies on a smooth horizontal table. Two men, each of mass
 M 
2, are standing on the edges of two perpendicular radii at
 A
 and
 B
.
AR C P B 
 Find the displacement of the centre of the disc if (a) The two men walk radically relative to the disc so as to meet at the centre
C 
. (b) The two men walk along the circumference to meet at the midpoint(P) of the are
 AB.
 
Q. 104. There particles
 A
,
 B
 and
C 
 have masses
m
, 2
m
 and
m
 respectively. They lie on a smooth horizontal table connected by light inextensible strings
 AB
 and
 BC 
. The string are taut and <
 ABC 
 = 120°. An impulse is applied to particle
 A
 along
 BA
 so that it acquires a velocity
u
. Find the initial speeds of
 B
 and
C 
.
I 
A B 
120°
C 
L 
EVEL 
 3
 
Q. 105. A smooth hollow
U 
 shaped tube of mass 2
m
 is lying at rest on a smooth horizontal table. Two small balls of mass
m
, moving with velocity
u
 enter the tube simultaneously in symmetrical fashion. Assume all collisions to be elastic. Find the final velocity of the balls and the tube.
m m u u 
2m
 
Q. 106. There are 40 identical balls travelling along a straight line on a smooth horizontal table. All balls have equal speed
v
 and each one is travelling to right or left. All collisions between the balls is
5
 
 M
OMENTUM
 
 AND
 C
ENTER
 
OF
 M
 ASS
 
5.19
head on elastic. At some point in time all balls will have fallen off the table. The time at which this happens will definitely depend on initial positions of the balls. Over all possible initial positions of the balls; what is the longest amount of time that you would need to wait to ensure that the table has no more balls? Assume that length of the table is
 L
.
L
 Q. 107. A small ball of mass
m
 is suspended from the end
 A
 of a
 L
 shaped mass less rigid frame which is fixed to a block of mass
m
. The block is placed on a smooth table. The ball is given a horizontal impulse so as to impart it a velocity of
u
. The ball beings to rotate in a circle of radius
 R
about the point
 A
, while the block and the frame slide on the table. Find the tension in the string, to which the ball is attached, at the instant the ball is at the top most position. The rod does not interfere with the string during the motion.
R Au m m 
string
 
Q. 108. A heavy rope of mass
m
 and length 2
 L
 is hanged on a smooth little peg with equal lengths on two sides of the peg. Right part of the rope is pulled a little longer and released. The rope begins to slide under the action of gravity. There is a smooth cover on the peg (so that the rope passes through the narrow channel formed between the peg and the cover) to prevent the rope from whiplashing. (a) Calculate the speed of the rope as a function of its length (
 x 
) on the right side. (b) Differentiate the expression obtained in (a) to find the acceleration of the rope as a function of
 x 
. (c) Write the rate of change of momentum of the rope as a function of
 x 
. Take downward direction as positive (d) Find the force applied by the rope on the peg as a function of
 x 
. (e) For what value of
 x 
, the force found in (d) becomes zero? What will happen if there is no cover around the peg?
cover
Lx 
 
Q. 109. Two thin rings of slightly different radii are joined together to make a wheel (see figure) of radius
 R
. There is a very small smooth gap between the two ring. The wheel has a mass
 M
and its centre of mass is at its geometrical centre. The wheel stands on a smooth surface and a small particle of mass
m
 lies at the top (
 A
) in the gap between the rings. The system is released and the particle begins to slide down along the gap. Assume that the ring does not lose contact with the surface.
q 
 R B A
 (a) As the particle slides down from top point
 A
 to the bottom point
 B
, in which direction does the centre of the wheel move? (b) Find the speed of centre of the wheel when the particle just reaches the bottom point
 B
. How much force the particle is exerting on the wheel at this instant? (c) Find the speed of the centre of the wheel at the moment the position vector of the particle with respect to the centre of the wheel makes an angle
q 
 with the vertical. Do this calculation assuming that the particle is in contact with the inner ring at desired value of
q 
. Q. 110. A large number of small identical blocks, each of mass
m
, are placed on a smooth horizontal surface with distance between two successive blocks being
d 
. A constant force
F 
 is applied on the first block as shown in the figure.
d1 2 3 4 5F
4
 
 5.20
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
 (a) If the collisions are elastic, plot the variation of speed of block 1 with time. (b) Assuming the collisions to be perfectly inelastic, find the speed of the moving blocks after
n
 collisions. To what value does this speed tend to if
n
 is very large.
 
Q. 111. Two small balls, each of mass
m
 are placed on a smooth table, connected with a light string of length 2
l
, as shown in the figure. The midpoint of the string is pulled along
 y
 direction by applying a constant force
F 
. Find the relative speed of the two particles when they are about to collide. If the two masses collide and stick to each other, how much kinetic energy is lost.
l  l 
m m 
 
Q. 112. A block of mass
 M 
 is tied to a spring of force constant
K 
 and the system is suspended vertically. Consider three situations shown in fig. (a), (b) and (c). (a) In fig. (a), an insect of mass
 M 
 is clinging to the block and the system is in equilibrium. The insect leaves the block and falls. Find the amplitude of resulting oscillations. (b) In fig. (b), an insect of mass
 M 
 is resting on the top of the block and the system is in equilibrium. The insect suddenly jumps up with a sped
u g M K 
=
 and the block starts oscillating. Find amplitude of oscillation assuming that the insect never falls back on the block. (c) In fig. (c), an insect of mass
 M 
 falls on the block that is in equilibrium. The insect hits the block with velocity
u g M K 
=
 while moving downwards and sticks to the block. Find the amplitude of oscillation.
u(a) (b) (c)
 
Q. 113. A massive ball (
 A
) is dropped from height
h
 on a smooth horizontal floor.
 A
 smaller ball (
 B
) is also dropped simultaneously. Initially ball
 B
was just touching ball
 A
 (see fig.). Radii of both balls is much smaller than
h
. Ball
 A
 hits the floor, rebounds and immediately hits
 B
. Motion of both the balls is vertical before the collision of two balls. All collision are elastic and there is no friction. Ball
 B
 lands at point
P
 on the ground after colliding with
 A
. Find
OP
, assuming that it is large compared to radius of
 A
.
q q 
O AB 
 
Q. 114. Disc A of radius
 R
 is lying flat on a horizontal surface. Disc
 B
 is also at rest. Disc
C 
, which is identical to
 B
 is traveling along the surface with its velocity parallel to the line joining the centre
C 
1
and
C 
2
 of the discs
 A
 and
 B
. The distance between the line
C 
1
C 
2
 and the line of motion of centre of disc
C 
 is 3.
r 
, where
r 
 is radius of both
 B
and
C 
. Impact of
C 
 
with
 B
 is completely elastic. Subsequently it is observed that both
 B
and
C 
 just miss hitting the disc
 A
. Find the radius (
 R
) of
 A
 in terms of
r 
.
C
3
r 
BC
2
 C
1
A
 Q. 115. A mass
m
 moving with speed
u
in
 x 
 direction collides elastically with a stationary mass 2
m
. After the collision, it was found that both masses have equal
 x 
 components of velocity. What angle does the velocity of mass 2
m
 make with the
 x 
 axis? Q. 116. A ball of mass
 M
collides elastically with another stationary ball of mass
m
. If
 M
>
m
, find the maximum angle of deflection of
 M 
.
5
 
 M
OMENTUM
 
 AND
 C
ENTER
 
OF
 M
 ASS
 
5.21
 
Q. 117. A tennis ball is lying on a rigid floor. A steel ball is dropped on it from some height. The steel ball bounces vertically after hitting the ball on the floor. Is it possible that the tennis ball will also bounce?
1.
 
D
 p
 = = Ê Ë Áˆ ¯ ˜ 
-
1029329
1
 Ns;
 q 
cos
2.
 
F 
2
3.
 (a) 400
 J 
 (b) Both performed equal work.
4.
a i j
car 
 = − +
2 2ˆ ˆ
5,
 
734
2
-Ê Ë Áˆ ¯ ˜ 
 r 
 AV 
6.
 
m r 
min
 =+
( )
mgS V u
 2
7.
 
mu
2
8.
 (i)
 M m
=
1
 (ii) 1/3
9.
u V 
+
2
10.
 3.24 ms
–1
making an angle of 44° with the normal to the wall
11.
 A time
ug
 after first collision
12.
 
 M m
£
2
13.
 
V  K  M  x 
=
2
0
14.
 (a)
KE  mu
loss2
3(b) No
15.
 (a) 3
m/s
 towards left (b)
K 
min
 = 2
 J 
(c)
K 
min
= 2
 J 
16.
 
m ug M m
222
2
µ 
2
−
( )
ANSWERS
17.
 
u
2
18.
(a)
O DABXCt
 
O
 –
 A
 –
 D
"
 
 Ball
1. 
C 
 –
 A
 –
 B
 
"
 
 Ball
2. (b) 10
19.
 2
20.
 
u uV V 
1 22 1
−−
21.
 (a)
OV
0
V
0
V
1
V
2
 (b)
V
1
O V
2
V
0
V
0
5
 
 5.22
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
 (c)
O V
2
V
1
V
0
V
0
 
V 
1
 = 0,
V 
2
 =
V 
0
 (d)
V 
1
 = 0.5
m/s
 ;
V 
2
 = 1.5
m/s
; % loss in KE = 37.5%
22.
 (a) The heavier particle moves with a velocity of  
221
0
V 
 in the direction of
V 
0
. Lighter particle moves with velocity 1921
0
V 
opposite to
V 
0
 The incident particle comes to rest. The other particle moves with
V 
0
.
23.
 0.1
m
24.
less than
 H 
25.
 1/3
26.
Slightly less than 6
m/s
27.
 13
m
28.
 
n n
+
( )
12
29.
 100
m
, 92.8%
30.
 (a)
e
 = 0 (b)
13
32.
12
33.
 (a)
 
 A
 will be at rest and
 B
 will have a velocity
u 
(b) (c) Both will be travelling with velocity
u
2. Loss in
KE  mu
2
4
34.
 
 L
260
35.
 203
kg
36.
 (i) (ii) A particle next to the blank vertex.
37.
(a) 12
0
k m x 
 (b) 34
0
kx m
38.
 0.2
m
39.
 (a) 4
m/s
 (b) 1560
 N 
40.
 (To right)
41.
40.56
m
42.
t  MLF 
=
43
43.
 No
44.
 7.5
 R
 (a) 10
m/s
 (b) 100
m
46.
v u M bt 
=+
12
47.
 (i)
T 
s
1
 =
T 
s
2
 =
T 
s
3
 =
r 
 Av
2
 (ii)
p 
 MV mu
0
 
48.
(a)
 
5
s
 
(b) 25
ms
–1
49.
 
F 
 
~
 1200
 N 
 (a)
g
2
 (b)
gvl
−
020
51.
(a) Smaller bullet (b)
 Mu M Aut 
+
 r 
 (c)
 M  An Au M t 
r r 
󰁬
1
+ÈÎ͢˚˙
 
52.
 (a)
 
85
 Lg
 
(b)
 
mgL
5
53.
 (a)
 
23
mg
 
(b)
 
2
 gL
54.
 
∆
t s
=
112
4
 
 M
OMENTUM
 
 AND
 C
ENTER
 
OF
 M
 ASS
 
5.23
55.
(a)
d 
 = 90
 L
 
u gL
=
90 (b)
m gL
45
56.
 
V m s V m s V m s
 A B C 
367307337 /;/;/ 
57.
 (a)
 Lv
 (b)
34
 
mv
2
58.
 40
m/s
60.
 (i)
 
 L
5
 
(ii)
 
5
m/s
–1
61.
232
 L H g
+ÈÎÍ͢˚˙˙
m 
62.
 5
m/s
63.
(a) 53
m
 (b) 9691067
=
.
 m
65.
T 
P
 
=
v
2
;
F 
ext 
 =
v
2
66.
 (i)
v Mv M x 
=+
0
λ 
 (ii)
T  M v M x 
=+
3320
λ λ 
67.
 (i) (a) 16
0224
k  m M  M m
+
 (b)
m  M k 
3
 (ii)
k 
10
 =
k 
1
68.
 4
69.
 (a) Left (b) Zero
70.
 2.96
m
 (a) 90
kg
 (b)
14
73.
 (b)
hv eg
=−
( )
222
12cos
 θ
 
 x e vg
= +
( )
122
2
sin
 q 
74.
 (a) 916 (b) 36 %
75.
 120°
76.
 (a)
q 
0
 = 60° (b)
T  mg
=
23
77.
 (i)
v m M  M mu v mu M m
m M 
= −+     =+    
;2
 (ii)
 Mg R r  R r 
+−      
78.
 
41115
2
sinsinsin
q q q q 
gR
 -
( )
+
where sin =
79.
 (i)
 H  L
54
 (ii) (a) 34
mg
 (b)
54
l
 (c) 234
π
lg
 (d) 52
lg
81.
 (b)
u u uu
2020
33
82.
 (a) 2
 R
 + 9
h
83.
 6
m/s
84.
 
mx  M 
85.
 23
u
, 180°
86.
 
Ft ml
2
42
+
5
 
 5.24
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
87.
 
m
2
= 4
kg
 ;
l
0
 = 6
cm
88.
(c) 12
1202
m m V 
+
( )
89.
v k m x 
r 
 =
32
90.
23
mk  v
0
91.
 (a) A circle of radius
mR M m
+
 (b)
m M 
=
1
92.
 
2
 2
π=
 MV  R
93.
 1.92 J
94.
 
1115
h
 (a) Closer to O (b) 0.03
 R
96.
 
mgK 
3
97.
 (a)
 R
sin
αα
 (b)
V V 
cm
=−
( )
=−
( )
2214221gR; gR
π π π
 (c)
 R
π
98.
 (a)
t  RV 
00
2
=π
 (b)
V 
0
2
99.
 23
0
 ML M M 
+
( )
100.
 (a)
mu M m
+
 (b)
mg
.
u
101.
 (a) 6
cm
 (b) 0.18
m/s
102.
 (a) No (b) 85
m
103.
 (a)
 R
22 (b) 2122
−
( )
 R
104.
v u v u
 B c
23111411,
105.
 
v
ball
 = 0;
v
tube
 =
u
106.
 Lv
107.
T  mu Rmg
= −
2
13
108.
 (a)
v g L x L
= −
( )
 (b)
a g L x L
= −
( )
 (c)
dpdt mg x L L
= -Ê Ë Á ˆ ¯ ˜ 
2
2
 (d)
F mg x L L
= − −      
12
2
 (e)
 x L L
= +
2
109.
 (a) First moves to right and then to left (b)
v m gR M M m
w
 =+
( )
2 (c)
vm gR M m Mm
w
 = -
( )
+
( )
 +
21
2222
coscosco
q q q 
110.
(a) See the solution for the graph (b)
nnFd mFd m
+
1;
111.
2
 FlmFl
;
112.
(a)
 MgK 
 (b)
2
 MgK 
 (c) 62
 MgK 
113.
162122
h
sincos
q q 
+ÈÎ͢˚˙
 
114.
 R r 
= +−     
3131
115.
45°
116.
sin
-1
 (
m/M 
)
117.
 Yes.
5
 
L 
EVEL 
 1
 
Q. 1. The pulley of radius
 R
 can rotate freely about its axle as shown in the figure. A thread is tightly wrapped around the pulley and its free end carries a block of mass
m.
 When the block is at a height
h
above the ground the system is released (i.e., the pulley is made free to rotate & the block is allowed to fall) and at the same instant the axle is moved up keeping it horizontal all the time. When the block hits the floor the axle has gone up by a distance 2
h
. Find the angle by which the pulley must have rotated by this time.
h R 
 
Q. 2. A disc is rolling without sliding on a horizontal surface. Velocity of the centre of the disc is
v.
 Find the maximum relative speed of any point on the circumference of the disc with respect to point
P.
v P 
30°
ROTATIONAL MOTION
 
Q. 3. A ring is rolling, without slipping on a horizontal surface with constant velocity. Speed of point
 A
 (at the top) is
v
 A
. After an interval
T 
, the speed of point
 A
 again becomes
v
 A
.
 During what fraction of the interval
T 
 speed of point
 A
 was greater than
32
v
 A
.
A 
 
Q. 4. Calculate the ratio of moment of inertia of a thin uniform disc about axis 1 and 2 marked in the figure.
O
 is the centre of the disc.
30°45°
O 
1
 
Q. 5. A uniform circular disc has a sector of angle 90° removed from it. Mass of the remaining disc is
 M 
. Write the moment of inertia of the remaining disc about the axis
 xx 
 shown in figure (Radius is
 R
)
x x O R 
06
4
 
 6.2
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
 
Q. 6. An Indian bread ‘‘Roti’’ is a uniform disc of mass
 M 
 and radius
 R
. Before eating a person usually folds it about its diameter (say about
 x 
 axis). After folding it a sector of angle 60° is removed from it. Find the moment of inertia of the remaining ‘‘Roti’’ about Z-axis.
O Z X O Z X 
60°
 
Q. 7. A uniform rectangular plate has side length
 and
2
. The plate is in
 x 
 –
 y
 plane with its centre at origin and sides parallel to
 x 
 and
 y
 axes. The moment of inertia of the plate about an axis passing through a vertex (say A) perpendicular to the plane of the figure is
 I 
0
. Now the axis is shifted parallel to itself so that moment of inertia about it still remains
 I 
0
.Write the locus of point of intersection of the axis with
 xy
 plane.
O
y Al 
2
l x 
 
Q. 8. A thin semi circular cylindrical shell has mass
 M
and radius
 R
. Find its moment of inertia about a line passing through its centre of mass parallel to the axis (shown in figure) of the cylinder.
axis
R 
 
Q. 9. Consider a uniform square plate shown in the figure.
 I 
1
,
 I 
2
,
 I 
3
 and
 I 
4
 are moment of inertia of the plate about the axes 1, 2, 3 and 4 respectively. Axes 1 and 2 are diagonals and 3 and 4 are lines passing through centre parallel to sides of the square. The moment of inertia of the plate about an axis passing through centre and perpendicular to the plane of the figure is equal to which of the followings. (a)
 I 
3
 +
 I 
4
(b)
 I 
1
 +
 I 
3
(c)
 I 
2
 +
 I 
3
(d) 12
1234
 I I I I 
+ + +
( )
2413O
 
Q. 10. An asteroid in the shape of a uniform sphere encounters cosmic dust. A thin uniform layer of dust gets deposited on it and its mass increases by 2%. Find percentage change in its moment of inertia about diameter.
 
Q. 11. (i) Consider an infinitesimally thin triangular strip having mass
 M 
 and length
 L
. Find the moment of inertia of the strip about on axis passing through its tip and perpendicular to the plane. Compare the result with moment of inertia of a uniform disc of mass
 M 
 and radius
 L
 about an axis passing through its centre and perpendicular to the plane the disc. Why the two expressions are same?
L 
 (ii) A circular fan made of paper is in shape of a disc of radius
 R
. The fan can be folded (various stages shown in figure (
a
) through (
 f 
)) to the shape of a thin stick. The moment of inertia of the circular fan about an axis passing through centre
O
 and perpendicular to the plane of the figure is
 I MR
02
12
=
 where
 M 
 = mass of the fan.
a  b 
(a)
O 
(b)
a  b O R R a  b 
(c)
O 
(c)
4
 
 R
OTATIONAL
 M
OTION
 
6.3
b R a O 
(d)
O R a  b 
(e)
O R a b 
(f)
 (a) How does the moment of inertia (
 I 
), about an axis perpendicular to the plane of the figure passing through
O
, change as the fan is folded through stage a to b to c to d to e? (b) When the fan is completely folded in the shape of a stick (fig. (
 f 
)), write its moment of inertia about the above mentioned axis. Note : Moment of inertia of a uniform rod about an axis through its end and perpendicular to it is 
 ML
2
3
.
 
Q. 12. A uniform rectangular plate has moment of inertia about its longer side, equal to
 I.
 The moment of inertia of the plate about an axis in its plane, passing through the centre and parallel to the shorter sides is also equal to
 I 
. Find its moment of inertia about an axis passing through its centre and perpendicular to its plane.
 
Q. 13. A uniform rectangular plate has been bent as shown in the figure. The two angled parts of the plate are of identical size. The moment of inertia of the bent plate about axis
 xx 
 is
 I 
. Find its moment of inertia about an axis parallel to
 xx
and passing through the centre of mass of the plate.
x x D E O B b AC 
60°
 
Q. 14. Three identical rings each of mass
 M 
 and radius
 R
 are welded together with their planes mutually perpendicular to each other. Ring
 A
 is vertical and
 B
 is also vertical in a plane perpendicular to
 A
. Ring
C 
 is in horizontal plane. Find moment of Inertia of this system about a horizontal axis perpendicular to the plane of the figure passing through point
P
 (top point of ring
 A
)
AB C 
 
Q. 15. Determine the moment of inertia of the shaded area about
 y
 axis. The mass of the shaded area is
 M 
. 
y Lx h 
1
 h 
2
 
Q. 16. Two uniform semicircular discs, each of radius
 R
, are stuck together to form a disc. Masses of the two semicircular parts are
 M 
 and 3
 M.
 Find the moment of inertia of the circular disc about an axis perpendicular to its plane and passing through its centre of mass.
3
M M 
 
Q. 17. A stick
 AB
 of mass
 M 
 is tied at one end to a light string
OA
. A horizontal force
F 
 =
 Mg
 is applied at end
 B
 of the stick and its remains in equilibrium in position shown. Calculate angles
a 
 and
b 
. 
O 
b a 
AB F 
4
 
 6.4
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
 
Q. 18. When brakes are applied on a moving car, the car dips to the front. Why ? [That is try to show that front wheels are more pressed as compared to rear ones when the brakes are applied]. Assume that centre of mass of the car is equidistant from the front and rear wheels.
 
Q. 19. A uniform wire has been bent in shape of a semi circle. The semicircle is suspended about a horizontal axis passing through one of its ends, so that the semicircular wire can swing in vertical plane. Find the angle
a 
 that the diameter of the semicircle makes with vertical in equilibrium.
a 
AB 
 
Q. 20. A uniform cylindrical body of radius
r 
 has a conical nose. The length of the cylindrical and conical parts are 4
r 
 and 3
r 
 respectively. Mass of the conical part is
 M 
. The body rests on a horizontal surface as shown. A ring of radius
r 
2 is to be tightly fitted on the nose of the body. What is maximum permissible mass of the ring so that the body does not topple?
r r r 
 Q. 21. There is a step of height
h
 on an incline plane. The step prevents a ball of radius
 R
 from rolling down. (a) If the inclination (
q 
) of the incline is increased gradually, at what value of
q 
the ball will just manage to climb the step?
q 
   h
R 
 (b) Does the gravitational potential energy of the ball increases or decreases as it climbs the step? Q. 22. The centre of mass of an inhomogeneous sphere is at a distance of 0.3
 R
 from its geometrical centre.
 R
is the radius of the sphere. Find the maximum inclination (
q 
) of an incline plane on which this sphere can be placed in equilibrium. Assume that friction is large enough to prevent slipping. Q. 23. Rectangular block
 B
, having height
h
 and width
d 
 has been placed on another block
 A
 as shown in the figure. Both blocks have equal mass and there is no friction between
 A
 and the horizontal surface.
 A
 horizontal time dependent force
F
=
kt 
 is applied on the block
 A
. At what time will block
 B
 topple? Assume that friction between the two blocks is large enough to prevent
 B
 from slipping.
d h B A F 
 Q. 24. A cylinder
C 
 rests on a horizontal surface. A small particle of mass
m
 is held in equilibrium connected to an overhanging string as shown. The other end of the mass less string is being pulled horizontally by a force
F 
 as shown. Find
F 
.
C AF 
 Q. 25. A hollow cylindrical pipe of mass
 M 
 and radius
 R
 has a thin rod of mass
m
 welded inside it, along its length.
 A
 light thread is tightly wound on the surface of the pipe. A mass
m
0
 is attached to the end of the thread as shown in figure. The system stays in equilibrium when the cylinder is placed such that
a 
 = 30°. The pulley shown in figure is a disc of mass
 M 
2. (a) Find the direction and magnitude of friction force acting on the cylinder.
4
 
 R
OTATIONAL
 M
OTION
 
6.5
 (b) Express mass of the rod ‘
m
’ in terms of
m
0
M m  R 
HorizontalRough table
 Am 
o
 Q. 26. A sphere of radius
 R
 is supported by a rope attached to the wall. The rope makes an angle
q 
 = 45° with respect to the wall. The point where the rope is attached to the wall is at a distance of 32
 R
 from the point where the sphere touches the wall. Find the minimum coefficient of friction ( between the wall and the sphere for this equilibrium to be possible.
R 
3 /2
R 
string45°
 Q. 27. A uniform rod has mass
 M 
 and length 4
 L.
 It rests in equilibrium with one end on a rough horizontal surface at
 A
. At point
 B
, at a distance 3
 L
 from
 A
, it is supported by a fixed smooth roller. The rod just remains in equilibrium when
q 
 = 30° (a) Find the normal force applied by the horizontal surface on the rod at point
 A
. (b) Find the coefficient of friction between the rod and the surface.
q 
   3
   L
  4
   L
AB 
 Q. 28. A wheel is mounted on frictionless central pivot and it can rotate freely in the vertical plane. There is a horizontal light rod fixed to the wheel below the pivot. There is a small sleeve of mass
m
 which can slide along the rod without friction. The sleeve is connected to a light spring. The other end of the spring is fixed to the rim as shown. The sleeve is at the centre of the rod and the spring is relaxed. Now the wheel is held at rest and the sleeve is moved towards left so as to compress the sprig by some distance. The sleeve and the wheel are released simultaneously from this position. (a) Is it possible that the wheel does not rotate as the sleeve perform SHM on the rod ? (b) Find the value of spring constant
k 
 for situation described in (a) to be possible. The distance of rod from centre of the wheel is
d.
PivotWheelSleeveSpringRod
O d 
 Q. 29. A string is wrapped around a cylinder of mass
 M 
 and radius
 R
. The string is pulled vertically upward to prevent the centre of mass from falling as the string unwinds. Assume that the cylinder remains horizontal throughout and the thread does not slip. Find the length of the string unwound when the cylinder has reached an angular speed
w 
.
a 
T Mg 
 Q. 30. A mass less string is wrapped around a uniform disc of mass
m
 and radius
r 
. The string passes over a mass less pulley and is tied to a block of mass
 M
at its other end (see figure). The system is released from rest. Assume that the string does not slip with respect to the disc.
4
 
 6.6
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
 (a) Find the acceleration of the block for the case
 M 
 =
m
r 
Disc
m M 
 (b) Find
 M m
 for which the block can accelerate upwards. Q. 31. A solid uniform sphere of mass
 M 
 and radius
 R
 can rotate about a fixed vertical axis. There is no frictional torque acting at the axis of rotation. A light string is wrapped around the equator of the sphere. The string has exactly 6 turns on the sphere. The string passes over a light pulley and carries a small mass
m
 at its end (see figure). The string between the sphere and the pulley is always horizontal. The system is released from rest and the small mass falls down vertically. The string does not slip on the sphere till 5 turns get unwound. As soon as 5
th
 turn gets unwound completely, the friction between the sphere and the string vanishes all of a sudden. (a) Find the angular speed of the sphere as the string leaves it. (b) Find the change in acceleration of the small mass
m
after 5 turns get unwound from the sphere.
m M 
 Q. 32. A disc of mass
m
 and radius
 R
 lies flat on a smooth horizontal table. A mass less string runs halfway around it as shown in figure. One end of the string is attached to a small body of mass
m
 and the other end is being pulled with a force
F 
. The circumference of the disc is sufficiently rough so that the string does not slip over it. Find acceleration of the small body.
R m F m 
 Q. 33. A uniform quarter circular thin rod of mass
 M 
 and radius
 R
is pivoted at a point
 B
 on the floor. It can rotate freely in the vertical plane about
 B
. It is supported by a smooth vertical wall at its other free end
 A
 so that it remains at rest. Find the reaction force of wall on the rod.
B R A
 Q. 34. A ball is rolling without sliding down an incline. Is the force applied by the ball on the incline larger than or less than its (ball’s) own weight ? Q. 35. A solid sphere of mass
 M
and radius
 R
 is covered with a thin shell of mass
 M 
. There is no friction between the inner wall of the shell and the sphere. The ball is released from rest, and then it rolls without slipping down an incline that is inclined at an angle
q 
 to the horizontal. Find the acceleration of the ball. Q. 36. A homogeneous solid sphere of radius
 R
 is resting on a horizontal surface. It is set in motion by a horizontal impulse imparted to it at a height
h
 above the centre. If
h
 is greater than
h
0
, the velocity of the sphere increases in the direction of its motion after the start. If
h
 <
h
0
, the velocity decreases after the start. Find
h
0
h 
4
 
 R
OTATIONAL
 M
OTION
 
6.7
 Q. 37. A boy pushes a cylinder of mass
 M 
 with the help of a plank of mass
m
 as shown in figure. There is no slipping at any contact. The horizontal component of the force applied by the boy on the plank is
F 
. Find (a) The acceleration of the centre of the cylinder (b) The friction force between the plank and the cylinder
M F m 
 Q. 38. (i) A solid sphere of radius
 R
 is released on a rough horizontal surface with its top point having thrice the velocity of its bottom point A (
V 
 A
 =
V 
0
) as shown in figure. Calculate the linear velocity of the centre of the sphere when it starts pure rolling.
AB V = V 
B 
 3
 0
V =V 
A
 0
 (ii) Solid sphere of radius
 R
is placed on a rough horizontal surface with its centre having velocity
V 
0
 towards right and its angular velocity being
w 
0
(in anticlockwise sense). Find the required relationship between
V 
0
 and
w 
0
 so that -
w 
0
R V 
0
the slipping ceases before the sphere loses all its linear momentum.the sphere comes to a permanent rest after some time.the velocity of centre becomes zero before the spinning ceases. Q. 39. A thin pencil of mass
 M 
 and length
 L
 is being moved in a plane so that its centre (i.e. centre of mass) goes in a circular path of radius
 R
 at a constant angular speed
w 
. However, the orientation of the pencil does not change in space. Its tip (A) always remains above the other end (B) in the figure shown (a) Write the kinetic energy of the pencil. (b) Find the magnitude of net force acting on the pencil.
R AB 
 Q. 40. In figure (a) there is a uniform cylinder of mass
 M 
 and radius
 R
. Length of the cylinder is
 L
 = 3
 R
. The cylinder is rolling without sliding on a horizontal surface with its centre moving at speed
V 
. In figure (b) the same cylinder is moving on a horizontal surface with its centre moving at speed
V
and the cylinder rotating about a vertical axis passing through its centre. [Place your pencil on the table and give a sharp blow at its end. Look at the motion of the pencil. This is how the cylinder is moving]. The angular speed is
ω 
 =
V  R
. Write the kinetic energy of the cylinder in two cases. In which case, the kinetic energy would have been higher if length of the cylinder were doubled (= 23
 R
).
C C  V 
V
w w 
(a) (b)
 Q. 41. There is a fixed hollow cylinder having smooth inner surface. Radius of the cylinder is
 R
 = 4
m
. A uniform rod of
 M 
 = 4
kg
 and length
 L
 = 4
m
 is released from vertical position inside the cylinder as shown in the figure. Convince yourself that the rod will perform pure rotation about the axis of the cylinder passing through
O
.
4
 
 6.8
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
Fid the angular speed of the rod when its becomes horizontal.
O
 Q. 42. A disc shaped body has two tight windings of light threads - one on the inner rim of radius
 R
 = 1
m
 and the other on outer rim of radius 2
 R
 (see figure). It is kept on a horizontal surface and the ends of the two threads are pulled horizontally in opposite directions with force of equal magnitude
F 
 = 20
 N.
 Mass of the body and its moment of inertia about an axis through centre
O
and perpendicular to the plane of the figure are
 M 
 = 4
kg
 and
 I 
 = 8
kg
–
m
2
 respectively. Find the kinetic energy of the body 2 seconds after the forces begin to act, if  (i) the surface is smooth, (ii) the surface is rough enough to ensure rolling without sliding.
R 
2
R 
 O
F F 
 Q. 43. A uniform square plate has mass
 M 
 and side length
a
. It is made to oscillate in vertical plane in two different ways shown in figure (A) and (B). In figure (A), the plate is hinged at its upper corners with the help of two mass less rigid rods each of length
a
. The rods can rotate freely about both ends.
fig. (a) fig. (b)
aaaa
 In figure (B) the plate is rigidly connected at the centre of its top edge to a mass less rod of length
a
. The rod can rotate about its upper end only. In both cases the plate is pushed from its equilibrium position so that centre of mass of the plate acquires a speed
V 
. In which case will the centre of mass of the plate rise to a greater height. There is no friction Q. 44. A thin carpet of mass 2
m
 is rolled over a hollow cylinder of mass
m.
 The cylinder wall is thin and radius of the cylinder is
 R
. The carpet rolled over it has outer radius 2
 R
 (see figure). This roll is placed on a rough horizontal surface and given gentle push so that the carpet begins to roll and unwind. Friction is large enough to prevent any slipping of the carpet on the floor. Also assume that the carpet does not slip on the surface of the cylinder. The entire carpet is laid out on the floor and the hollow cylinder rolls out with speed
V 
. Find
V 
.
R 2R 
 Q. 45. A uniform rod of mass
 M 
 is moving in a plane and has a kinetic energy of 43
2
 MV 
 where
V 
 is speed of its centre of mass. Find the maximum and minimum possible speed of the end point of the rod. Q. 46. The propeller of a small airplane is mounted in the front. The propeller rotates clockwise if seen from behind by the pilot. The plane is flying horizontally and the pilot suddenly turns it to the right. Will the body of the plane have a tendency to get inclined to the horizontal? If yes, does the nose of the plane veer upward or downward? Why? Q. 47. A massive star is spinning about its diameter with an angular speed
w  p 
0
1000
=
rad/day. After its fuel is exhausted, the star collapses under its own gravity to form a neutron star. Assume that the volume of the star decreases to 10
–12
 times the original volume and its shape remains spherical. Assuming that density of the star is uniform, find the angular speed of the neutron star.
4
 
 R
OTATIONAL
 M
OTION
 
6.9
 Q. 48. A square plate of side length 2
m
 has a groove made in the shape of two quarter circles joining at the centre of the plate. The plate is free to rotate about vertical axis passing through its centre. The moment of inertia of the plate about this axis is 4
kg
 –
m
2.
A small block of mass 1
kg
 enters the groove at end A travelling with a velocity of 2
m/s
 parallel to the side of the square plate. The block move along the frictionless groove of the horizontal plate and comes out at the other end
 B
 with speed
V 
. Find
V 
 assuming that width of the groove is negligible.
2
m A
2
m/s B 
 Q. 49. A disc of mass
m
 and radius
 R
 lies flat on a smooth horizontal table. A particle of mass
m
, moving horizontally along the table, strikes the disc with velocity
V 
 while moving along a line at a distance 
 R
2from the centre. Find the angular velocity acquired by the disc if the particle comes to rest after the impact.
V  R 
 /2
m 
 Q. 50. A disc of mass
 M 
 and radius
 R
 is rotating with angular velocity
w 
0
 about a vertical axis passing through its centre (O). A man of mass
 M 
2 and height
 R
2 is standing on the periphery. The man gradually lies down on the disc such that his head is at a distance
 R
2 from the centre and his feet touching the edge of the disc. For simplicity assume that the man can be modelled as a thin rod of length
 R
2. Calculate the angular speed (
w 
) of the platform after the man lies down.
O R 
 /2
w 
O R R 
 /2
w 
0 
 Q. 51. A uniform block of mass
 M 
 and dimensions as shown in the figure is placed on a rough horizontal surface and given a velocity
V 
0
 to the right.
 A
 is a point on the surface to the left of the block. (a) Write the angular momentum of the block about point
 A
 just after it begins to move (b) Due to friction the block stops. What happened to its angular momentum about point
 A
? Which torque is responsible for change in angular momentum of the block?
b 
a
V 
0 
A
 Q. 52. ABCFED is a uniform plate (shown in figure).
 ABC
and
 DEF 
 are circular arcs with common centre at
O
 and having radii
a
 and 2
a
respectively. This plate is lying on a smooth horizontal table. A particle of mass half the mass of the plate strikes the plate at point
 A
 while travelling horizontally along the
 x 
 direction with velocity
u
. The particle hits the plate and rebounds along negative
 x 
 with velocity
u
2. Find the velocity of point
 D
 of the plate immediately after the impact. [Take 2891
π
]
B E AD O C  F x 
a a
5
 
 6.10
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
 Q. 53. A uniform rod of mass
m
 and length
 L
 is fixed to an axis, making an angle
q 
 with it as shown in the figure. The rod is rotated about this axis so that the free end of the rod moves with a uniform speed ‘
v
’. Find the angular momentum of the rod about the axis. Is the angular momentum of the rod about point A constant?
q 
B A
 Q. 54. A mass
m
 in attached to a mass less string and swings in a horizontal circle, forming a conical pendulum, as shown in the figure. The other end of the string passes through a hole in the table and is dragged slowly so as to reduce the length
l
. The string is slowly drawn up so that the depth
h
 shown in the figure becomes half. By what factor does the radius (
r 
) of the circular path of the mass
m
 change?
O h  l r m 
L 
EVEL 
 2
 Q. 55. A flat rigid body is moving in
 x 
 –
 y
 plane on a table. The plane of the body lies in the
 x 
 –
 y
 plane. At an instant it was found that some of the velocity components of its three particles
 A
,
 B
 and
C 
 were
V 
 Ax 
 = 4
m/s
,
V 
 Bx 
 = 3
m/s
 and
V 
cy
 = – 2
m/s
, respectively. At the instant the three particles
 A
,
 B
 and
C 
 were located at (0,0) (3,4) , (4,3) (all in meter ) respectively in a co-ordinate system attached to the table. (a) Find the velocity of
 A
,
 B
 and
C 
 (b) Find the angular velocity of the body.
y B C x A
 Q. 56. A wheel is rolling without sliding on a horizontal surface. Prove that velocities of all points on the circumference of the wheel are directed towards the top most point of the wheel.
P  V 
P 
 Q. 57. There is a fixed half cylinder of radius
 R
 on a horizontal table. A uniform rod of length 2
 R
leans against it as shown. At the instant shown,
q 
 = 30° and the right end of the rod is sliding with velocity
v
.
R O V 
 (a) Calculate the angular speed of the rod at this instant. (b) Find the vertical component of the velocity of the centre of the rod at this instant. Q. 58. A disc of radius
 R
 is rolling without sliding on a horizontal surface at a constant speed of
v
AR 
 /2
B R V 
4
 
 R
OTATIONAL
 M
OTION
 
6.11
 (a) What is speed of points
 A
 and
 B
 on the vertical diameter of the disc ? Given
 AB R
2 (b) After what time, for the first time, speed of point
 A
becomes equal to present speed (i.e., the speed at the instant shown in the figure) of point
 B
? Q. 59. A uniform disc of radius
 R
 = 23
m
 is moving on a horizontal surface without slipping. At some instant its angular velocity is
w 
 = 1 rad/s and angular acceleration is
a 
 = 3
 rad/s
2
.
AC P r B 
 (a) Find acceleration of the top point
 A
. (b) Find acceleration of contact point
 B
. (c) Find co - oridnates (
r 
,
q 
) for a point
P
 which has zero acceleration. Q. 60. A convex surface has a uniform radius of curvature equal to 5
 R
. A wheel of radius
 R
 is rolling without sliding on it with a constant speed
v
. Find the acceleration of the point (
P
) of the wheel which is in contact with the convex surface.
R V P 5R 
 Q. 61.
 AB
 is a non uniform plank of length
 L
 = 4
 R
 with its centre of mass at
C 
 such that
 AC 
 =
 R
. It is placed on a step with its one end
 A
 supported by a cylinder of radius
 R
 as shown in figure. The centre of mass of the plank is just outside the edge of the step. The cylinder is slowly rolled on the lower step such that there is no slipping at any of its contacts. Calculate the distance through which the centre of the cylinder moves before the plank loses contact with the horizontal surface of the upper step.
B  C  AR 
 Q. 62. A wheel of radius
 R
 is rolling without sliding uniformly on a horizontal surface. Find the radius of curvature of the path of a point on its circumference when it is at highest point in its path. Q. 63. A wall is inclined to a horizontal surface at an angle of 120° as shown. A rod
 AB
 of length
 L
 = 0.75
m
 is sliding with its two ends
 A
 and
 B
on the horizontal surface and on the wall respectively. At the moment angle
q 
 = 20° (see figure), the velocity of end
 A
 is
v
 A
 = 1.5
m/s
 towards right. Calculate the angular speed of the rod at this instant. [Take cos 40° = 0.766]
q 
B L
120°
AV 
A
 Q. 64. In the figure the plank resting on two cylinders is horizontal. The plank is pulled to the right such that the centre of smaller cylinder moves with a constant velocity
v
. Friction is large enough to prevent slipping at all surfaces. Find- (a) The velocity of the centre of larger cylinder. (b) The ratio of accelerations of the points of contact of the two cylinders with the plank.
2
R R  V 
 Q. 65. A wire of linear mass density (
kg
 /m) is bent into
q 
0
q 
0
M O 
4
 
 6.12
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
 an arc of a circle of radius
 R
 subtending an angle 2
q 
 at the centre. Calculate the moment o inertia of this circular arc about an axis passing through its midpoint (
 M 
) and perpendicular to its plane. Q. 66. A metallic plate has been fabricated by welding two semicircular discs -
 D
1
 and
 D
2
 of radii
 R
 and 
 R
2 respectively (fig. a).
O
 and
O
' are the centre of curvature of the two discs and each disc has a mass 6
m
. The plate is in
 xy
 plane. Now the plate is folded along the
 y
 axis so as to bring the part
OAB
in
 yz
 plane. (fig. b). The plate is now set free to be able to rotate freely about the
 z
 – axis. A particle of mass
m
, moving with a velocity
v
 in the
 xy
 plane along the line
 x 
 =
d 
 hits the plate and sticks to it (
d 
 <
 R
). Just before collision speed of the particle was
v
.
Fig (b)Fig (a)
B R AO´ C E F  y x D 
1
D 
2
O y B A O C E  D 
2
x d m  v F Z 
 (a) Find the moment of inertia of inertia of the plate about
 z
axis. (b) Find the angular speed of the plate after collision. Q. 67. There is a square plate of side length
a
. It is divided into nine identical squares each of side 
a
3 and the central square is removed (see fig. (i)). Now each of the remaining eight squares of side length
a
3 are divided into nine identical squares and central square is removed from each of them (see fig. (ii)).
Fig. (ii)Fig. (i)
a
 Mass of the plate with one big and eight small holes is
 M 
. Find its moment of inertia about an axis passing through its centre and perpendicular to its plane. Q. 68.
 ABC 
 is an isosceles triangle right angled at
 A
. Mass of the triangular plate is
 M 
 and its equal sides are of length
a
. Find the moment of inertia of this plate about an axis through
 A
 perpendicular to the plane of the plate. Use the expression of moment of inertia for a square plate that you might have studied.
a
C A
a
 Q. 69. The triangular plate described in the last question has angle <
 A
 =
q 
 Now find its moment of inertia about an axis through
 A
 perpendicular to the plane of the plate.
q 
aa
B C A
 Q. 70. A thin uniform spherical shell of radius
 R
 is bored such that the axis of the boring rod passes through the centre of the sphere. The boring rod is a cylinder of radius
 R
2. Take the mass of the sphere before boring to be
 M 
. (a) Find the mass of the leftover part (b) Find the moment of inertia of the leftover part about the axis shown.
axis
R 
 /2
R 
4
 
 R
OTATIONAL
 M
OTION
 
6.13
 Q. 71. Consider an equilateral prism as shown in the figure. The mass of the prism is
 M 
 and length of each side of its cross section is
a.
 Find the moment of inertia of such a prism about the central axis shown.
axis 
a
 Q. 72. In the arrangement shown in figure the cylinder of mass
 M 
 is at rest on an incline. The string between the cylinder and the pulley (P) is horizontal. Find the minimum coefficient of friction between the incline and the cylinder which can keep the system in equilibrium. Also find the mass of the block. Assume no friction between the pulley (P) and the string.
Block
q 
P M 
 Q. 73. A horizontal stick of mass
m
 has its right end attached to a pivot on a wall, while its left end rests on the top of a cylinder of mass
m
which in turn rests on an incline plane inclined at an angle
q 
. The stick remains horizontal. The coefficient of friction between the cylinder and both the plane and the stick is
m 
. Find the minimum value of
m 
 
as function of
q 
 for which the system stays in equilibrium.
q 
 Q. 74. Consider the object shown in the figure. It consist of a solid hemisphere of mass
 M 
 and radius
 R
. There is a solid rod welded at its centre. The object is placed on a flat surface so that the rod is vertical. Mass of the rod per unit length is
 M  R
2. What is the maximum length of the rod that can be welded so that the object can perform oscillations about the position shown in diagram? Note : Centre of mass of a solid hemisphere is at a distance of 38
 R
 from its base.
R  
 Q. 75. Two cylinders
 A
 and
 B
 have been placed in contact on an incline. They remain in equilibrium. The dimensions of the two cylinders are same. Which cylinder has larger mass?
q 
  A  B
 Q. 76. The ladder shown in the figure is light and stands on a frictionless horizontal surface. Arms
 AB
 and
 BC 
 are of equal length and
 M 
 and
 N
are their mid points. Length of
 MN 
 is half that of
 AB
. A man of mass
 M 
 is standing at the midpoint of
 BM 
. Find the tension in the mass less rod
 MN 
. Consider the man to be a point object.
B M  N A C 
 Q. 77. A uniform metal sheet of mass
 M 
 has been folded to give it
 L
 shape and it is placed on a rough floor as shown in figure. Wind is blowing horizontally and hits the vertical face of the sheet as shown. The speed of air varies linearly from zero at floor level to
v
0
 at height
 L
 from the floor. Density of air is
r 
. Find maximum value of
v
0
 for which the sheet will not topple. Assume that air particles striking the sheet come to rest after collision, and that the friction is large enough to prevent the
4
 
 6.14
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
sheet from sliding.
Wind
LL L
 Q. 78. Three identical cylinders have mass
 M 
 each and are placed as shown in the figure. The system is in equilibrium and there is no contact between
 B
 and
C 
. Find the normal contact force between
 A
 and
 B
.
AB  C 
 Q. 79. A spool is kept in equilibrium on an incline plane as shown in figure. The inner and outer radii of the spool are in ratio
r  R
=
12. The force applied on the thread (wrapped on part of radius
r 
) is horizontal. Find the angle that the force applied by the incline on the spool makes with the vertical. [Take tan
−
       °
1
3519
]
q 
=60°
F R r 
 Q. 80. A uniform hemisphere placed on an incline is on verge of sliding. The coefficient of friction between the hemisphere and the incline is
m 
= 0.3. Find the angle
f 
 that the circular base of the hemisphere makes with the horizontal. Given sin (tan
–1
 0.3) 0.29 and sin
–1
 (0.77) 50°
q f 
 Q. 81.
 A L
 shaped, uniform rod has its two arms of length
l
 and 2
l
. It is placed on a horizontal table and a string is tied at the bend. The string is pulled horizontally so that the rod slides with constant speed. Find the angle
q 
 that the longer side makes with the string. Assume that the rod exerts uniform pressure at all points on the table.
q 
String2
l l 
 Q. 82. A uniform meter stick
 AB
 of mass
 M 
 is lying in state of rest on a rough horizontal plane. A small block of mass
m
 is placed on it at its centre
C.
 A variable force
F 
 is applied at the end
 B
 of the stick so as to rotate the stick slowly about
 A
 in vertical plane. The force
F 
 always remains perpendicular to the length of the stick. The stick is raised to
q 
 and it was observed that neither the end
 A
 slipped on the ground nor the block of mass
m
 slipped on the stick.
q 
F A C  B F B m 
 
q q q q 
 Q. 83. A ladder of mass
 M 
 and length
 L
 stays at rest against a smooth wall. The coefficient of friction between the ground and the ladder is
m 
.
4
 
 R
OTATIONAL
 M
OTION
 
6.15
 (a) Let
F 
wall
,
W 
 and
F 
g
 be the force applied by wall, weight of the ladder and force applied by ground on the ladder. Argue to show that the line of action of these three forces must intersect. (b) Using the result obtained in (a) show that line of action of
F 
g
 makes an angle tan
–1
 (2 tan
q 
) with the horizontal ground where
q 
 is the angle made by the ladder with the ground. (c) Find the smallest angle that the ladder can make with the ground and not slip. (d) You climb up the ladder, your presence makes the ladder more likely to slip. Where are you at
 A
 or
 B
?
C 
 is the centre of mass of the ladder.
q 
AB C 
 Q. 84. A uniform rod
 AB
 has mass
 M 
 and length
 L.
It is in equilibrium supported in vertical plane by three identical springs as shown in figure. The springs are connected at
 A
,
C 
 and
 D
 such that 
 AC CD L
= =
3. Assume that the springs are very stiff and the angle
a 
 made by the rod with the horizontal in equilibrium position is very small. (All springs are nearly vertical). Calculate the tension in the three springs.
a 
L / 
 33 2 1
B D C AL / 
 3
 Q. 85. A uniform rod of length
b
 can be balanced as shown in figure. The lower end of the rod is resting against a vertical wall. The coefficient of friction between the rod and the wall and that between the rod and the support at A is
m 
. Distance of support from the wall is
a
. (a) Find the ratio
ab
 if the maximum value of
q 
 is
q 
. (b) Find the ratio
ab
 if the minimum value of
q 
 is
q 
.
q 
A 
a
 Q. 86. A uniform rectangular block is moving to the right on a rough horizontal floor (the block is retarding due to friction). The length of the block is
 L
 and its height is
h
. A small particle (
 A
) of mass equal to that of the block is stuck at the upper left edge. Coefficient of friction between the block and the floor is
m 
 =
23. Find the value of
h
 (in terms of  
 L
) if the normal reaction of the floor on the block effectively passes through the geometrical centre (
C 
) of the block.
AC 
 
Lh 
 Q. 87. A uniform cubical block of mass
 M 
 and side length
 L
 is lying on the edge of a rough table with 
14
th
 of its edge overhanging. When a small block  of mass
m
 is placed on its top surface at the right edge (see fig.), the cube is on verge of toppling. The block of mass
m
 is given a sharp horizontal impulse so that it acquires a velocity towards
 B
. The small block moves on the top surface and falls on the other side. What is maximum coefficient of friction between the small block and the cube so that the cube does not rotate as the block moves over it. Assume that the friction between the cube and the table is large enough to prevent sliding of the cube on the table.
B m ALM L / 
 4
3
 
 6.16
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
 Q. 88. A uniform rod of mass
 M 
 and length
 L
 is hinged at its end to a wall so that it can rotate freely in a horizontal plane. When the rod is perpendicular to the wall a constant force
F 
 starts acting at the centre of the rod in a horizontal direction perpendicular to the rod. The force remains parallel to its original direction and acts at the centre of the rod as the rod rotates. (Neglect gravity). (a) With what angular speed will the rod hit the wall ? (b) At what angle
q 
 (see figure) the hinge force will make a 45° angle with the rod ?
q 
F 
wall
F 
 Q. 89. A rod of mass
 M 
 = 5
kg
 and length
 L
 = 1.5
m
 is held vertical on a table as shown. A gentle push is given to it and it starts falling. Friction is large enough to prevent end
 A
 from slipping on the table.
B M LA
q q 
 (a) Find the sum of linear momentum of all the particles of the rod when it rotates through an angle
q 
= 37°. (b) Find the friction force and normal reaction force by the table on the rod, when
q 
= 37°. (c) Find value of angle
q 
 when the friction force becomes zero. 
[tan/]3734
∞
= and = 10
g m s
 Q. 90. (a) In the system shown in figure 1, the uniform rod of length
 L
 and mass
m
 is free to rotate in vertical plane about point
O
. The string an pulley are mass less. The block has mass equal to that of the rod. Find the acceleration of the block immediately after the system is released with rod in horizontal position. (b) System shown in figure 2 is similar to that in figure 1 apart from the fact that rod is mass less and a block of mass
m
 is attached to the centre of the rod with the help of a thread. Find the acceleration of both the blocks immediately after the system is released with rod in horizontal position.
Fig.(1)
LO m m 
Fig.(2)
m  m L
 Q. 91.
Q AP B 
 A light thread is wrapped tightly a few turns around a disc
P
 of mass
 M 
. One end of the thread is fixed to the ceiling at
 B
. The other end of the thread is passed over a mass less pulley (
Q
) and carries a block of mass
 M 
. All segment of the thread (apart from that on the pulley and disc) are vertical when the system is released. Find the acceleration of block
 A
. On which object – the block A or the ceiling at
 B
 – does the thread exert more force ? Q. 92. An equilateral triangle is made from three mass less rods, each of length
l
. Two point masses m are attached to two vertices. The third vertex is hinged and triangle can swing freely in a vertical plane as shown. It is released the position shown with one of the rods vertical. Immediately after the system is released, find – (a) tensions in all three rods (specify tension or compression), (b) accelerations of the two masses
4
 
 R
OTATIONAL
 M
OTION
 
6.17
pivot
m m l l l 
 Q. 93.
 A B
 A rod of mass
 M 
 and length
 L
 is hinged about its end
 A
 so that it can rotate in vertical plane. When the rod is released from horizontal position it takes
t 
0
 time for it to become vertical. (a) A particle of mass
 M 
 is stuck at the end
 B
 of the rod and the rod is once again released from its horizontal position. Will it take more time or less time (than
t 
0
) for the rod to become vertical from its horizontal position. (b) At what distance
 x 
 from end
 A
 shall the particle of mass
 M 
 be stuck so that it takes minimum time for the rod to become vertical from its horizontal position. Q. 94. A disc is free to rotate about an axis passing through its centre and perpendicular to its plane. The moment of inertia of the disc about its rotation axis is
 I 
. A light ribbon is tightly wrapped over it in multiple layers. The end of the ribbon is pulled out at a constant speed of u. Let the radius of the ribboned disc be
 R
 at any time and thickness of the ribbon be
d 
 (<<
 R
). Find the force (
F 
) required to pull the ribbon as a function of radius
 R
.
d u R 
 Q. 95. A uniform rod of mass
 M 
 and length
 L
 is hinged at its lower end on a table. The rod can rotate freely in vertical plane and there is no friction at the hinge. A ball of mass
 M 
 and radius
 R L
=
3 is placed in contact with the vertical rod and a horizontal force
F 
 is applied at the upper end of the rod. (a) Find the acceleration of the ball immediately after the force starts acting.
F  
 (b) Find the horizontal component of hinge force acting on the rod immediately after force
F 
 starts acting. Q. 96.
R A
q 
 A ring of mass
 M 
 and radius
 R
 is held at rest on a rough horizontal surface. A rod of mass
 M 
 and length
 L
 = 23
 R
is pivoted at its end A on the horizontal surface and is supported by the ring. There is no friction between the ring and the rod. The ring is released from this position. Find the acceleration of the ring immediately after the release if
q 
 = 60°. Assume that friction between the ring on the horizontal surface is large enough to prevent slipping of the ring. Q. 97. A uniform semicircular wire is hinged at ‘
 A
’ so that it can rotate freely in vertical plane about a horizontal axis through ‘
 A
’. The semicircle is released from rest when its diameter
 AB
 is horizontal.
A B 
 Find the hinge force at ‘
 A
’ immediately after the wire is released. Q. 98. A uniform solid hemisphere
 A
 of mass
 M 
 radius
 R
 is joined with a thin uniform hemispherical
5
 
 6.18
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
shell
 B
 of mass
 M 
 and radius
 R
 (see fig.). The sphere thus formed is placed inside a fixed box as shown. The floor , as well as walls of the box are smooth. On slight disturbance, the sphere begins to rotate. Find its maximum angular speed (
w 
0
) and maximum angular acceleration (
a 
0
) during the subsequent motion. Do the walls of the box apply any force on the sphere while it rotates?
B A
 Q. 99.
r O 
  R
 In the arrangement shown, the double pulley has a mass
 M 
 and the two mass less threads have been tightly wound on the inner (radius =
r 
) and outer circumference (radius
 R
 = 2
r 
). The block shown has a mass 4
 M 
. The moment of inertia of the double pulley system about a horizontal axis passing through its centre and perpendicular to the plane of the figure is
 I  Mr 
2
2
. (a) Find the acceleration of the center of the pulley after the system is released. (b) Two seconds after the start of the motion the string holding the block breaks. How long after this the pulley will stop ascending? Q. 100.
 
A thread is tightly wrapped on two pulleys as shown in figure. Both the pulleys are uniform disc with upper one having mass
 M 
 and radius
 R
 being free to rotate about its central horizontal axis. The lower pulley has mass
m
 and radius
r 
 and it is released from rest. It spins and falls down. At the instant of release a small mark (
 A
) was at the top point of the lower pulley. (a) After what minimum time (
t 
0
) the mark will again be at the top of the lower pulley?
M m R Ar 
 (b) Find acceleration of the mark at time
t 
0
. (c) Is there any difference in magnitude of acceleration of the mark and that of a point located on the circumference at diametrically opposite end of the pulley. Q. 101. A point mass
m
 = 1
kg
 is attached to a point P on the circumference of a uniform ring of mass
 M 
 = 3
kg
and radius
 R
 = 2.0
m
. The ring is placed on a horizontal surface and is released from rest with line OP in horizontal position (O is centre of the ring). Friction is large enough to prevent sliding. Calculate the following quantities immediately after the ring is released- (a) angular acceleration (
a 
) of the ring, (b) normal reaction of the horizontal surface on the ring and (c) the friction force applied by the surface on the ring. [Take g = 10
m/s
2
]
O R P 
 Q. 102. A light thread has been tightly wrapped around a disc of mass
 M 
 and radius
 R
. The disc has been placed on a smooth table, lying flat as shown.
 
R 
M m 
5
 
 R
OTATIONAL
 M
OTION
 
6.19
 The other end of the string has been attached to a mass
m
 as shown. The system is released from rest. If
m
 =
 M 
, which point of the disc will have zero acceleration, immediately after the system is released? Q. 103. A spool has the shape shown in figure. Radii of inner and outer cylinders are
 R
 and 2
 R
 respectively. Mass of the spool is 3
m
 and its moment of inertia about the shown axis is 2
mR
2
. Light threads are tightly wrapped on both the cylindrical parts. The spool is placed on a rough surface with two masses
m
1
 =
m
 and
m
2
 = 2
m
connected to the strings as shown. The string segment between spool and the pulleys
P
1
 and
P
2
 are horizontal. The centre of mass of the spool is at its geometrical centre. System is released from rest. (a) What is minimum value of coefficient of friction between the spool and the table so that it does not slip? (b) Find the speed of
m
1
 when the spool completes one rotation about its centre.
R 
2R 
axis
R 2R m 
2 
m 
1
P 
1
P 
2
 Q. 104. 
q 
F F B Ah 
 A heavy uniform log of mass
 M 
 is pulled up an incline surface with the help of two parallel ropes as shown in figure. The ropes are secured at point
 A
 and
 B
. The height of the incline is h and its inclination is
q 
. (a) Find the minimum force
F 
0
 needed to roll the log up the incline. (b) Find the work done by the force in moving the log from the bottom to the top of the incline if the applied force is
F 
 = 2
F 
0
 Q. 105. In the figure shown, the light thread is tightly wrapped on the cylinder and masses of plank and cylinder are same each equal to
m
. An external agent begins to pull the plank to the right with a constant force
F 
. The friction between the plank and the cylinder is large enough to prevent slipping. Assume that the length of the plank is quite large and the cylinder does not fall off it for the time duration concerned. (a) Find the acceleration of the cylinder. (Hint : don’t write any equations) (b) Find the kinetic energy of the system after time
t 
.
R F 
 Q. 106. A disc of radius
r 
 = 0.1
m
 is rolled from a point
 A
 on a track as shown in the figure. The part
 AB
 of the track is a semi-circle of radius
 R
 in a vertical plane. The disc rolls without sliding and leaves contact with the track at its highest point
 B
. Flying through the air it strikes the ground at point
C 
. The velocity of the center of mass of the disc makes an angle of 30° below the horizontal at the time of striking the ground. At the same instant, velocity of the topmost point
P
 of the disc is found to be 6
m/s
. (Take
g
 = 10
m/s
2
).
P C 
30°
AB 
 (a) Find the value of
 R
. (b) Find the velocity of the center of mass of the disc when it strikes the ground. (c) Find distance
 AC 
.
5
 
 6.20
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
 Q. 107. A trough has two identical inclined segments and a horizontal segment. A ball is released on the top of one inclined part and it oscillates inside the trough. Friction is large enough to prevent slipping of the ball. Time period of oscillation is
T 
. Now the liner dimension of each part of the trough is enlarged four times. Find the new time period of oscillation of the ball. Q. 108. A uniform cylinder is lying on a rough sheet of paper as shown in fig. The strip is pulled horizontally to the right with a constant acceleration of
a
0
 = 6
m/s
2
. Initially the cylinder is located at a distance of
 L
 = 10
m
 from the left end of the strip. Find the velocity of the centre of the cylinder at the instant it moves off the edge of the strip. Assume that the cylinder does not slip.
L = 10 m 
a
0 
= 6 m/s 
2 
 Q. 109.
m M F 
 A hollow pipe of mass
 M
= 6
kg
 rests on a plate of mass
m
= 1.5
kg
 . The thickness of the pipe is negligible. The coefficient of friction at all contacts is
m 
 = 0.2. The system is initially at rest. A horizontal force F of magnitude 25
 N 
 is applied on the plate as shown in figure. Will the cylinder slide on the plate? Find the acceleration of the centre of the cylinder. Q. 110.
k AB k 
 A hollow cylindrical pipe A has mass
 M 
 and radius
 R
. With the help of two identical springs (each of force constant k) it is connected to solid cylinder B having mass
 M 
 and radius
 R
. The springs are connected symmetrically to the axle of the cylinders. Moment of inertia of the two Bodies A and B about their axles are
 I 
 A
 =
 MR
2
 and
 I MR
 B
 =
12
2
 respectively. Cylinders are pulled apart so as to stretch the springs by
 x 
0
 and released. During subsequent motion the cylinders do not slip. (a) Find acceleration of the centre of mass of the system immediately after it is released. (b) Find the distance travelled by cylinder A by the time it comes to rest for the first time after being released. Q. 111. Two identical uniform thin rods have been connected at right angles to form a ‘
T 
’ shape. One end of a rod is connected to the centre of the other rod. Length of each rod is
 L
. The upside down ‘
T 
’ can swing like a pendulum about a horizontal axis passing through the top end (see fig.). Axis is perpendicular to plane of the fig. The speed of the meeting point of the two rods is
u
= 2
 gL
 when it is at its lowest position. Calculate the angular acceleration of the ‘
T 
’ shaped object when it is at extreme position of its oscillation. Q. 112. A uniform rod of mass
 M 
 and length
 L
 is hinged at its lower end so as to rotate freely in the vertical plane of the fig. There is a small tight fitting bead of mass
 M 
6 on the rod at a distance
34
 L
 from the hinged end. A small mass less pin welded to the rod supports the bead. The system is released from the vertical position shown. It was observed that the bead just begins to slide on the rod when the rod becomes horizontal. (a) Find the normal contact force between the rod and the bead when the rod gets horizontal. What is the direction of this force?
3
L
4
M 
6
5
 
 R
OTATIONAL
 M
OTION
 
6.21
 (b) Find the coefficient of friction between the bead and rod. Q. 113
cm
L
 Two astronauts having mass of 75
kg
 and 50
kg
 are connected by a rope of length
 L
 = 10
m
 and negligible mass. They are in space, orbiting their centre of mass at an angular speed of
w 
0
 = 5
rad 
 / 
s
. The centre of mass itself is moving uniformly in space at a velocity of 10
m/s
. By pulling on the rope, the astronauts shorten the distance between them to
 Lm
25. How much work is done by the astronauts in shortening the distance between them ? Assuming that the astronauts are athletic and each of them can generate a power of 500 watt, is it possible for the two astronauts to reduce the distance between them to 5
m
, within a minute? Q. 114. In the figure shown a plank of mass
m
 is lying at rest on a smooth horizontal surface. A cylinder of same mass
m
 and radius
r 
 is rotated to an angular speed
w
0
 and then gently placed on the plank. It is found that by the time the slipping between the plank and the cylinder cease, 50% of total kinetic energy of the cylinder and plank system is lost. Assume that plank is long enough and
m 
 is the coefficient of friction between the cylinder and the plank.
m, r 
w 
0 
m 
 (a) Find the final velocity of the plank. (b) Calculate the magnitude of the change in angular momentum of the cylinder about its centre of mass. (c) Distance moved by the plank by the time slipping ceases between cylinder and plank. Q. 115. A horizontal turn table of mass 90
kg
is free to rotate about a vertical axis passing through its centre. Two men – 1 and 2 of mass 50
kg
 and 60
kg
 respectively are standing at diametrically opposite point on the table. The two men start moving towards each other with same speed (relative to the table) along the circumference. Find the angle rotated by table by the time the two men meet. Treat the men as point masses.
1 2 
 Q. 116. A
 L
 shaped uniform rod has both its sides of length
l
. Mass of each side is
m
. The rod is placed on a smooth horizontal surface with its side AB horizontal and side BC vertical. It tumbles down from this unstable position and falls on the surface. Find the speed with which end C of the rod hits the surface.
B Al l C 
 Q.117. 
V
 A flat horizontal belt is running at a constant speed
V 
. There is a uniform solid cylinder of mass
 M 
 which can rotate freely about an axle passing through its centre and parallel to its length. Holding the axle parallel to the width of the belt, the cylinder is lowered on to the belt. The cylinder begins to rotate about its axle and eventually stops slipping. The cylinder is, however, not allowed to move forward by keeping its axle fixed. Assume that the moment of inertia of the cylinder about its axle is 12
2
 MR
 where
 M 
 is its mass and
 R
 its
5
 
 6.22
P
ROBLEMS
 
IN
 P
HYSICS
 
FOR
 
JEE
 A
DVANCED
 radius and also assume that the belt continues to move at constant speed. No vertical force is applied on the axle of the cylinder while holding it. (a) Calculate the extra power that the motor driving the belt has to spend while the cylinder gains rotational speed. Assume coefficient of friction =
m 
. (b) Prove that 50% of the extra work done by the motor after the cylinder is placed over it, is dissipated as heat due to friction between the belt and the cylinder.
 
Q.118. A uniform disc of mass
 M 
 and radius
 R
 is rotating freely about its central vertical axis with angular speed
w 
0
. Another disc of mass
m
 and radius
r 
 is free to rotate about a horizontal rod
 AB
. Length of the rod
 AB
 is
 L
 (<
 R
) and its end A is rigidly attached to the vertical axis of the first disc. The disc of mass m, initially at rest, is placed gently on the disc of mass
 M 
 as shown in figure. Find the time after which the slipping between the two discs will cease. Assume that normal reaction between the two discs is equal to mg. Coefficient of friction between the two discs is
m 
.
R  ALB 
w 
0
 
Q.119. P is a fixed smooth cylinder of radius
 R
 and
Q
 is a disc of mass
 M 
 and radius
 R
. A light thread is tightly wound on
Q
 and its end is connected to a rope ABC. The rope has a mass m and length 
π
 R
2and is initially placed on the cylinder with its end A at the top. The system is released from rest. The rope slides down the cylinder as the disc rolls without slipping. The initial separation between the disc and the cylinder was
 L R
=π
2 (see fig). Find the speed with which the disc will hit the cylinder. Assume that the rope either remains on the cylinder or remains vertical; it does not fly off the cylinder.
fixed
Q R 
Thread
O AB C P L
rope
R 
 
Q.120. A uniform cube of mass
 M 
 and side length
a
 is placed at rest at the edge of a table. With half of the cube overhanging from the table, the cube begins to roll off the edge. There is sufficient friction at the edge so that the cube does not slip at the edge of the table. Find - (a) the angle
q 
 through which the cube rotates before it leaves contact with the table. (b) the speed of the centre of the cube at the instant it breaks off the table. (c) the rotational kinetic energy of the cube at the instant its face AB becomes horizontal.
C B D A P D B C A
q 
0
 
Q.121. A uniform frictionless ring of mass
 M 
 and radius
 R
, stands vertically on the ground.
 A
 wall touches the ring on the left and another wall of height
 R
 touches the ring on right (see figure).There is a small bead of mass m positioned at the top of the ring. The bead is given a gentle push and it being to slide down the ring as shown. All surfaces are frictionless.
m M A
q 
R 
 (a) As the bead slides, up to what value of angle
q 
 the force applied by the ground on the ring is larger than
 Mg
?
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